Apparatus and method using measurements taken while drilling cement to obtain absolute values of mechanical rock properties along a borehole

ABSTRACT

An innovative apparatus and computer implemented methods to obtain values for a set of scalars corresponding to each force and displacement, which may be obtained from acoustical signals captured by sensors of a drill bit while drilling, in a material of known mechanical properties, such as a cement from casing the well, such that the application and use of the scalars in relation to measurements of the mechanics while drilling, such as the acceleration of the bit and motion of the bit captured by sensors such as accelerometers, allow for absolute values of mechanical rock properties to be obtained in rock formations, being drilled through, with otherwise unknown mechanical properties prior to drilling.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to and is a continuation-in-part applicationclaiming priority to co-pending U.S. patent application Ser. No.15/616,742 entitled “APPARATUS AND METHOD USING MEASUREMENTS TAKEN WHILEDRILLING CEMENT TO OBTAIN ABSOLUTE VALUES OF MECHANICAL ROCK PROPERTIESALONG A BOREHOLE,” filed Jun. 7, 2017, which claims the benefit ofpriority under 35 U.S.C. § 119(e) to U.S. Provisional Application No.62/346,894 titled “APPARATUS AND METHOD OF USING MEASUREMENTS TAKENWHILE DRILLING CEMENT TO OBTAIN ABSOLUTE VALUES OF MECHANICAL ROCKPROPERTIES ALONG A BOREHOLE,” filed Jun. 7, 2016, both of which arehereby incorporated by reference in their entirety.

This application is also related to and is a continuation-in-part ofco-pending U.S. patent application Ser. No. 14/850,710 entitled“APPARATUS AND METHOD USING MEASUREMENTS TAKEN WHILE DRILLING TO MAPMECHANICAL BOUNDARIES AND MECHANICAL ROCK PROPERTIES ALONG A BOREHOLE,”filed Sep. 10, 2015, which claims priority under 35 U.S.C. § 119 fromU.S. Provisional Application No. 62/048,669 entitled “APPARATUS ANDMETHOD USING MEASUREMENTS TAKEN WHILE DRILLING TO MAP MECHANICALBOUNDARIES AND MECHANICAL ROCK PROPERTIES ALONG A BOREHOLE,” filed onSep. 10, 2014, both of which are hereby incorporated by reference intheir entirety.

TECHNICAL FIELD

The present disclosure involves measurement while drilling (MWD)techniques that provide mechanical rock properties, and from whichfractures and other mechanical boundaries may be identified and used toimprove drilling and completion practices, among other things.

BACKGROUND AND INTRODUCTION

A method to obtain absolute values of mechanical rock properties fromMWD data using the forces on the bit and the motions of the bit whileinteracting with a rock formation with known values of mechanical rockproperties involves (i) informing the terms and loading conditions of astress-strain relationship with respect to the material symmetry of therock formation in relation to the orientation of the drilling well,involves (ii) obtaining values for a set of scalars with respect to themotion of the bit and the acceleration or forces on the bit in relationto the known absolute values of the mechanical rock properties, and(iii) in one particular example obtaining values for the set of scalarswhen the rock formation is cement.

When specified in absolute values, elastic coefficients can be used, ingeneral, to describe the deformation of a rock formation in response tothe forces acting on the rock formation, and in particular, to predictthe deformation of a hydrocarbon bearing formation in response to theforces acting on the formation when the forces are fluid pressuresgenerated during the emplacement of hydraulic fractures in connectionwith a hydraulic fracture stimulation treatment along a horizontal well.Reservoir models that are used to describe the deformation of areservoir as a result of hydraulic fracture emplacement require absolutevalues of the mechanical rock properties of the hydrocarbon bearing rockformation. Absolute values of mechanical rock properties are importantin order to provide meaningful results to plan a wellbore stimulationtreatment.

SUMMARY

Aspects of the present disclosure involve a method of characterizingrock properties while drilling comprising: receiving acoustical signalsobtained from one or more sensors positioned on a component of a bottomhole assembly. The sensors (e.g., accelerometers or strain gauges) maybe in operable communication with at least one data memory to store theacoustical signals, may perform some real-time processing. The datamemory may be on or in the bottom hole assembly, or the acoustical datamay be transmitted to a memory structure on the surface, in two possibleexamples. The acoustical signals, which may also be consideredvibrations, are generated from a drill bit interacting with a material,such as cement, having a known mechanical rock property first and thenseparately with a rock formation, having to be determined rockproperties while drilling a wellbore. The method involves obtainingscalars by processing the acoustical signals and applying force oracceleration data, and displacement to a stress strain relationship forthe material with known rock properties, and then the method furtherinvolves processing the acoustical signals, from the drill bitinteracting with the rock formation, to obtain at least one set of datavalues representative of a mechanical rock property of the rockformation along the wellbore created by the drill bit interacting withthe rock formation for a period of time. Processing may be in real-timeor from stored data. The scalars are used to obtain absolute values orotherwise normalized values of the mechanical rock properties.

More specifically, the present disclosure describes an innovativetechnique to obtain values for a set of scalars corresponding to eachforce and displacement in a material of known mechanical properties suchthat the application and use of the scalars in relation to measurementsof the mechanics while drilling a well such as the acceleration of thebit and motion of the bit allow for absolute values to be obtained inrock formations with unknown mechanical properties.

In one possible example, the disclosure describes a method ofcharacterizing rock properties while drilling comprising receivingacoustical signals obtained from one or more sensors positioned on acomponent of a bottom hole assembly, the acoustical signals generatedfrom a drill bit interacting with a rock formation while drilling awellbore. The method further involves processing the acoustical signalsto obtain forces acting on the drill bit interacting with the rockformation while drilling the wellbore and to obtain displacements of thedrill bit interacting with the rock formation while drilling thewellbore, and scaling the forces and the displacements to obtaininformation representative of stresses and strains of the drill bitinteracting with the rock formation. The method then involves processingthe scaled forces and the displacements to obtain at least one set ofdata values representative of an absolute value of a mechanical rockproperty of the rock formation along the wellbore created by the drillbit interacting with the rock formation for a period of time.

In one example, the operation of scaling involves applying scalars tothe processed forces and displacements, the scalars derived fromacoustical signals generated from the drill bit acting on a sample, suchas cement, with known mechanical rock properties. The scalars may beobtained by receiving acoustical signals obtained from one or moresensors positioned on a component of a bottom hole assembly, theacoustical signals generated from a drill bit interacting with thesample. The scalars may further be obtained by processing the acousticalsignals to obtain forces acting on the drill bit interacting with thesample to obtain displacements of the drill bit interacting with thesample and processing the forces and the displacements to obtain atleast one set of scalars that conform to a stress strain relationship ofthe sample with known mechanical rock properties.

In one example, the stress strain relationship is:

${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$

where:

the a values are forces or accelerations acting on the drill bitinteracting with the sample;

the d values are displacements the drill bit interacting with thesample;

where C₁₁ and C₁₂ are populated with known rock properties of thesample; and

A-F are the at least one set of scalars being obtained.

In another example, the stress strain relationship is:

${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$

where:

the a values are forces or accelerations acting on the drill bitinteracting with the sample with respect to an axis of material symmetryof the sample;

the d values are displacements the drill bit interacting with the samplewith respect to the axis of material symmetry of the sample;

where the Cijs are populated with known rock properties of the sample;and

A-F are the at least one set of scalars being obtained.

In another aspect of the present disclosure, a method of calibratingmechanical rock property derivations from a drilling tool involvesobtaining a force data and a displacement data from signals obtainedfrom one or more sensors positioned on a component of a drilling toolproximate a drill bit, the force data and the displacement data being ofa drill bit interacting with a material with known mechanical rockproperties, and processing the force data and the displacement data toobtain a set of scalars that conform to a stress strain relationship ofthe material with a known mechanical rock property. In this example, thestress strain relationship may be:

${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$

where:

the a values are the force data with respect to an axis of materialsymmetry of the material;

the d values are the displacement data with respect to the axis ofmaterial symmetry of the material;

where C₁₁ and C₁₂ are populated with the known rock property of thematerial; and

A-F are the set of scalars.

In an alternative, the stress strain relationship may be:

${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$

where:

the a values are the force data with respect to an axis of materialsymmetry of the material;

the d values are the displacement data with respect to the axis ofmaterial symmetry of the material;

where the Cijs are populated with the known mechanical rock property ofthe material; and

A-F are the set of scalars.

These and other aspects of the present disclosure are set out below.

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1A-1B Illustrate reservoir-to-well connectivity where brittlerocks are generally associated with larger fracture creation and betterproppant support that is more permeable than ductile rock that producessmaller, less productive fractures which are prone to rapid compactionand closure and are less permeable.

FIG. 2A-2B is a diagram of a drill bit assembly including sensors formeasuring bit accelerations and forces on the bit, and which includes atleast one processing unit and tangible storage media in which to storeacceleration and/or force data, and which may also store processedacceleration and/or force data of the drill bits interaction with aformation while drilling.

FIGS. 3A-3C illustrate the cutting action of a drill bit, stick slipfracturing of a rock formation, torque on bit and weigh on bit forces,and related torque and displacement curves.

FIG. 4 illustrates root mean square computations for axial accelerationof the bit as used in computing rock properties.

FIG. 5A is a rotary displacement spectra obtained from measurement whiledrilling data.

FIG. 5B is a diagram illustrating lateral and rotary acceleration of thebit, while cutting, and useful in computing mechanical rock propertiesamong other advantages.

FIG. 6 is a diagram illustrating drill bit behaviour when encountering amechanical discontinuity or geological boundary, the cutting face of thedrill bit may change its orientation, as detectable from accelerationdata, in response to the orientation and stresses acting on theheterogeneity.

FIG. 7 is a graph of rock strength curves computed from torque andpenetration per revolution information based on measurement whiledrilling data.

FIG. 8A illustrates stress strain relationships based on an orientationof a well relative to a transverse isotropic axis of material symmetryand mechanisms whereby measurement while drilling techniques may be usedto compute elastic coefficients, from which temporal and/or spatialvariations in one or a combination of more of the measurements obtainedfrom the geophysical signal processing techniques are used to identifythe nature and occurrence of fractures, fracture swarms and othermechanical discontinuities (boundaries) such as bedding planes and/orfaults that offset or otherwise separate rock formations with differentmechanical rock properties.

FIG. 8B is a stress strain curve reflecting the relationships of FIG.8A.

FIG. 9A illustrates a constitutive stress strain relationship for anaxis of material symmetry parallel a well axis, and computations usingtorque on bit, weigh on bit and axial displacement to obtain elasticcoefficients from measurement of bit vibration and/or forces acting onthe bit.

FIG. 9B illustrates a constitutive stress strain relationship for anaxis of material symmetry perpendicular a well axis, and computationsusing torque on bit, weigh on bit and rotary displacement to obtainelastic coefficients from measurement of bit vibration and/or forcesacting on the bit.

FIGS. 10A-10C a constitutive stress strain relationships for an axis ofmaterial parallel and perpendicular a well axis, using accelerationdata.

FIG. 11 illustrates constitutive stress strain relationships for an axisof material symmetry parallel a well axis and axis of symmetryperpendicular a well axis, with Poisson's ration and Young's Modulus ofElasticity computed from measurement while drilling data of torque onbit, weigh on bit, and acceleration data to obtain lateral and axialdisplacement.

FIG. 12 illustrates stress strain curves obtained by the methodsdiscussed herein.

FIG. 13 illustrates two relative curves of Poisson's ration for avertical well, with cross over points identifying likely preexistingfractures in the fracture detection log.

FIG. 14A illustrates two relative curves of Young's Modulus ofElasticity for a horizontal well.

FIG. 14B illustrates two relative curves of Poisson's ration for thesame horizontal well, with cross over points identifying mechanical rockproperties indicative of preexisting fractures.

FIG. 15 If the forces acting on the formation in connection with thedrill bit and drilling fluid system when conducting drilling operationsare sufficient to overcome the failure criteria of a pre-existing fault,then the fault will slip or fail. Reactivation of a fault orpre-existing fracture can be evidenced by extracting a signal from thedrilling vibrations that is related to a microseismic event withattendant primary, compressional (P) and secondary, or shear (S)arrivals. In the case where the fault is perpendicular to the trajectoryof the wellbore the P-wave arrival is related the particle motionparallel to the axis of the drill string and the S- or transverse waveis the particle motion parallel to the lateral and torsional motion ofthe drill string. Deviations of the particle motion of the P- andS-waves relative to the orientation of the well can be used to determinethe orientation of the fault.

FIG. 16 is a flow diagram of a method of obtaining mechanical rockproperties of a formation proximate a well bore from measurements of bitbehavior taken while drilling.

FIG. 17 is a flow diagram depicting a method of obtaining scalarsaccording to one embodiment of the present disclosure.

FIG. 18 is a diagram illustrating stress and strain relationshipsdeveloped from acoustical signals captured by a three axis accelerometerpositioned on a measurement while drilling assembly of a drill.

FIG. 19 is a special purpose computer programmed with instructions toexecute methods discussed herein.

DETAILED DESCRIPTION

The present disclosure involves an inventive way of using drillingvibrations generated by the deformation of a rock formation in responseto forces acting on the rock formation, where the forces are related toa drill bit and drilling fluid system, to obtain mechanical rockproperties of a formation, identify the nature and occurrence offractures, fracture swarms and other mechanical discontinuities(boundaries) such as bedding planes and/or faults that offset orotherwise separate rock formations with different mechanical rockproperties.

As will be appreciated from the devices, systems and methods providedand disclosed herein, aspects of the present disclosure may also involvethe determination of absolute values of mechanical rock properties,which becomes possible from the application and use of (i) the forces oraccelerations of the drill bit and (ii) the displacements or motions ofthe drill bit such as those obtained from recording the near-bitmechanical drilling vibrations in relation to the drill bit interactingwith the rock formation. In one specific example, drilling vibrationsexperienced by the drill bit from its breaking of rock while drilling,propagate as acoustical signals that are translated into data byaccelerometers or other sensors positioned proximate the drill bit. Theacoustical signals are translated into mechanical rock propertiesaccording to the techniques discussed herein. Further, by first drillingthrough some known media, such as cement in the well, the system maycapture data and generate scalars that can be used to transform derivedmechanical rock properties, from data capture in the same manner, for anunknown media, such as a formation being drilled through, into absolutevalues of mechanical rock properties for that formation.

The techniques and measurements from this disclosure may be made, inwhole or in part, using downhole tools that are simple and rugged,allowing for a magnitude in order reduction in logging cost tocharacterize near-wellbore rock mechanical properties, which may furtherinclude identifying and characterizing intersected existing fracturelocations. The low cost to log a well, which may typically be less than0.5% of the total well cost, allows for widespread use of the technique.Detailed knowledge of rock property variability along a wellbore allowsfor grouping like-for-like rock types in variable length stages,avoiding losing reserves due to a lack of fracture initiation relativeto mixed rock strength stages. Scaling mechanical rock propertycomputations to further obtain absolute values of such mechanical rockproperty computations may further enhance the methods and systems.

Further elaboration of the method describes how mechanical rockproperties, and in particular elastic coefficients of a rock formation,can be determined through the application and use of innovative, newstress-strain relationships that systematically relate measurements ofthe forces acting on rock formation in connection with the drill bit anddrilling fluid system (stress) to the variations in the drillingvibrations generated by the deformation of the rock formation inresponse to the cutting action of the bit (strain).

The elastic coefficients can be used, in general to describe thedeformation of a rock formation in response to the forces acting on therock formation, and in particular, to predict the deformation of ahydrocarbon bearing formation in response to the forces acting on theformation where the forces are fluid pressures generated during theemplacement of hydraulic fractures in connection with a hydraulicfracture stimulation treatment along a horizontal well.

The processing of drilling vibrations when recorded using sensorsdeployed in a borehole in connection with a bottom hole assembly (BHA)according to the method disclosed here, can provide measurements ofmechanical rock properties including the nature and occurrence ofmechanical discontinuities, such as pre-existing fractures, which can beused to target sections of the well where the rock properties areconducive to economical hydraulic stimulation and to avoid sections thatare viewed as sub-commercial, where the rock properties are notconducive to economical hydraulic stimulation.

In another embodiment of the method, the elastic coefficients andvariations in the elastic coefficients that are obtained whileconducting drilling operations can be used for assisting, in real-time,the steering of the bottom hole assembly in order to maintain thetracking of the drill bit through geological formations as are targetedaccording to the desired mechanical rock properties, especially wherethe mechanical rock properties are relevant to the production ofcommercially significant hydrocarbons using hydraulic fracturingstimulation techniques.

The present disclosure generally relates to the production ofcommercially significant hydrocarbons from oilfield drilling operationsand completion operations. Recent unconventional resource developmenthas identified a need for economical determination of rock propertiesand natural fracture swarm locations along a horizontal well in order tooptimize the location and intensity of hydraulic stimulation treatments.

The techniques described in this disclosure will provide new informationfor selecting hydrocarbon bearing zones by differentiating betweenbrittle rocks generally associated with larger fracture creation andbetter proppant support that is more permeable than ductile rock thatproduces smaller, less productive fractures which are prone to rapidcompaction and closure and are less permeable. Natural fractureidentification also refines the process of hydraulic stimulationoptimization by providing direct measurement of zones that offer higherpermeability and higher hydrocarbon productivity.

Physical Basis of the Method

Aspects of the present disclosure involve methodologies that use broadband measurements (e.g., continuous, high resolution) of drillingvibrations and drilling dynamics data taken proximate the drill whileconducting drilling operations bit to log the mechanical properties of arock formation.

Drilling vibrations generated by the deformation and failure of a rockformation are generally related to the mechanical properties of the rockbeing drilled. It is generally understood that the depth of cut or thetooth penetration into the rock is inversely related to the strength ofthe rock. Higher amplitude drilling vibrations occur in rocks thatundergo a greater depth of cut and deeper tooth penetration in responseto the forces acting on the formation in connection with the drill bitand drilling fluid system, whereas lower amplitude drilling vibrationsoccur in rocks that undergo relatively lower depth of cut and lessertooth penetration. Increased depth of cut indicate the bit is movinginto an area of lesser relative mechanical rock strength, and decreasedrelative drilling vibrations indicate the bit is moving into an area ofgreater relative rock strength all other things being equal.

Generally speaking, rock formations that take a relatively long time todrill through or where the rate of penetration is slow are generallyreferred to strong or hard formations and have a lesser depth of cut inrelation to rock formations that are relatively weaker and less rigid.These basic principles have enabled the application and use oftechniques that take measurements of the hardness of a rock formation byforcing a tool into a rock to make an indentation where the depth of theindentation relative to the force applied is used to obtain a hardnesscharacteristic that is essentially a mechanical property of a rockformation.

The presence of mechanical discontinuities, such as pre-existingfractures, and geological boundaries, such as faults, in a rockformation generally act to weaken the rock formation. Fractured rockformations, are generally weaker and less rigid than intact, unfracturedor stiff or otherwise competent rock formations. As the drill bitencounters fractures in a rock formation the tooth penetration or depthof cut and subsequently the drilling vibrations will increase, becausethe rock formation is less rigid because it has been weakened by thepresence of fractures. Stated differently, as the drill moves into andthrough existing fractures, the measured mechanical rock strength willdecrease relative to the same rock formation without a fracture or withlesser fractures, for example.

General description of Fracture Identification in relation to the Method

Signal processing techniques are used to process the drilling data toidentify locations where the changes in the drilling vibration indicatethat the drill bit has encountered a mechanical discontinuity orgeological boundary. If the changes in the drilling vibration asexpressed through the results of the geophysical signal processingtechniques are rapid and discrete in both space and time, and thenreturn back to a long-term trend or the levels that were recorded priorto the change in the drilling vibrations, then it would indicate thatthat the drill bit has encountered and crossed a discrete mechanicaldiscontinuity because mechanical rock properties that are discrete inboth space and time are uniquely separated from the mechanicalproperties of a rock formation such as would be in the case of a drillbit penetrating a fracture face. If the changes in drilling vibrationare rapid and discrete and continue over a short interval, then thatwould indicate multiple fractures or a swarm of fractures has beenencountered.

If the signal processing techniques indicate that the changes in thedrilling vibration are rapid, but then do not revert back to the levelprior to the change and instead carry on at a new, significantlydifferent level, then that indicates a mechanical boundary where themechanical boundary that separates or offsets two different rockformations such as a bedding plane and or fault has been encountered andcrossed. Whether or not the boundary is related to a fault or a beddingplane depends on the inclination of the bit with respect to theorientation of the stratigraphy of the rock formation being drilled.Other information may also be used to determine whether or not themechanical boundary was a bedding-plane fault or bedding plane thatacted as a zone of weakness that had experienced measurable displacementin the past.

The description provides a method to evidence the presence of fractures,fracture swarms and other mechanical discontinuities such as faults andbedding planes that offset or otherwise separate rock formations withdifferent rock properties. The approach uses geophysical signalprocessing techniques that are sensitive to changes in the drillingvibrations where the changes are relative to some baseline, such as anormalized preceding set of drilling vibration data, and whether or notthe changes are discrete and then return back to the level prior to thechange or are maintained at a new level that is different than the levelobserved prior to the change.

The following method disclosed below further elaborates on the outlinedprinciples to provide a general, independent method to specify themechanical properties of a rock formation by processing the drillingvibrations in relation to the forces acting on the formation inconnection with a drill bit and, in some instances, a drilling fluidsystem, which includes the mud motor and the drilling fluid, includingmud, that turns the motor. This method specifies the mechanicalproperties of a rock formation through the application and use ofinnovative, new stress-strain relationships, among other advances.

Because the cutting depth or the penetration of the bit tooth into theformation is a measurement of displacement, the drilling vibrationsdescribe the strain experienced by a rock formation in response to thecutting action of the bit where greater cutting depths and a greaterpenetration of the bit tooth relative to the same volume of rock resultin a higher strain. Strain is understood to describe a change in volumeof a rock under some force. In some aspects, strain is reflected byaxial displacement of the bit per turn of the bit. In one example, axialdisplacement is computed as a double integral of the accelerometer data(axial) for one revolution of the bit to yield a distance measurementfor one revolution (or some other known number of turns). If this strainis calculated with respect to time, then the drilling vibrations can beused to describe the strain rate. The converse with regards to thestrain and the rate of strain is also held to be evident.

Through the techniques described herein, the drilling vibrationcharacteristics, which may be supplemented with drilling dynamics dataincluding forces on the bit, such as torque on bit and weight on bit,are translated into mechanical properties of a rock formation. The depthof cut, as obtained based on vibration assessment, may be normalizedagainst direct weight on bit and/or torque on bit measurements, orweight on bit or torque on bit measurements extrapolated from vibrationinformation.

General Stress-Strain Relationships

Stress-strain relationships are established by systematically relatingforces acting on the formation in connection with the drill bit anddrilling fluid system to the geophysical signal processing of drillingvibrations generated by the deformation of the rock in response to thecutting action of the bit. This approach allows elastic coefficients (K)to be derived in accordance with the following equation where (e) is thegeneral deformation (strain) of a rock formation in response to theforces acting on a rock formation (S) (stress).

S=K e

In accordance with some methods set out herein, strain (the motion ordisplacement of the bit is obtained by signal processing the drillingvibrations as are transmitted along the drilling assembly by sensorsdeployed in a borehole in connection with a BHA. Stress, in accordancewith some methods set out herein, is obtained from either (i) downholeor (ii) surface measurements of torque and/or weight on bit, or (iii) inanother example, the accelerations of the bit are related to forces onbit where it understood that the acceleration is a representation offorce per unit mass. It should be appreciated that forces can beconverted to stresses with knowledge of the effective contact area ofthe bit and formation, and the effective rock volume the bit is actingon. Conversely, forces can be substituted for stresses with theunderstanding that a geometric correction in relation to the effectivecontact area is required to obtain absolute values for the mechanicalrock properties. One example of such a contact area is the area of thebit.

The equations of linear elasticity are useful for describing therelationship between the changes in shape and position of a material inrelation to the forces acting on the material. Such stress-strainrelationships are known in general as Hooke's law where the coupling ofthe stress-strain relationship behavior is described through a matrix ofcoefficients whose values depend on the conditions used to load thematerial in relation to the structural symmetry of the material beingloaded. These coefficients are colloquially known as the cij's and canbe arranged in well-known and convenient forms to represent Young'sModulus of Elasticity (YME) and Poisson's Ratio (PR). In one example ofthe technique presented here, the YME and PR values are systematicallydetermined by the loading conditions of the bit in relation to the axisof material symmetry used to describe the rock formation.

In one specific implementation, the constitutive equations of linearelasticity are uniquely expressed through the application and use of MWDdata to (i) populate the variables of the constitutive equations oflinear elasticity and (ii) undertake an analysis of the constitutiveequations to obtain measurements of near-wellbore mechanical rockproperties of (a) Young's modulus of elasticity and (b) Poisson's ratio.Further, variations in the mechanical rock properties (e.g., YME and PR)are used to identify the nature and occurrence of mechanical boundariesor discontinuities in the subsurface such as fractures.

More specifically, a technique to determine near-wellbore mechanicalrock properties, YME and PR, from MWD data may involve processingmeasurements of the weight on bit (WOB), torque on bit (TOB), Annularfluid pressure (Ap), angular bit speed (RPM) and components of motiondescribing the acceleration of the bit, including axial, and the rotaryor tangential accelerations to (i) obtain sets of MWD data correspondingto known temporal and spatial positions along the borehole, (ii)calculate the forces acting on the rock formation in connection with thedrilling apparatus and drilling fluids, (iii) calculate thedisplacements of the bit as it is accommodated by the deformation of therock formation, (iv) inform the terms and loading conditions (variables)of a linear, elastic stress-strain relationship that describes theconstitutive behavior of the rock formation in relation to theorientation of the well and (v) using the constitutive linear elasticequations as determined through the application and use of the MWD datato calculate the aforementioned mechanical rock properties, (YME and PR)and (vi) analyzing the YME and PR with respect to the axis of materialsymmetry in relation to the orientation of the well to identify thenature and occurrence of mechanical boundaries and discontinuities suchas fractures and bedding planes among other things.

The present disclosure involves an innovative, new system, apparatus andmethod to specify, in general, the mechanical properties of a rockformation from an analysis of drilling vibrations generated by thecutting action of the bit and the deformation of the formation inresponse to the forces acting on the rock formation in connection withthe drill bit and drilling fluid system while conducting drillingoperations. Deformation may include elastic deformation, plasticdeformation, and failure of the rock, which may be consideredfracturing. Stated differently, aspects of the present disclosureinvolve obtaining information associated with the drilling of aborehole, while drilling, to identify mechanical rock properties of theformation being drilled. Such mechanical rock properties may be used, insome examples, to identify the presence of natural fractures or rockproperties more or less susceptible to stimulation techniques. Forexample, knowing mechanical rock properties along a borehole or thepresence of natural fractures along a borehole may be used to optimizehydraulic fracturing operations by focusing such fracturing on areaswhere it will be most effective, among other advantages. Mechanical rockproperties may include elastic coefficients (e.g., the cij's), strengthmeasurements such as initial yield strength, peak compressive strength,tensile strength YME, PR, shear modulus, bulk modulus, strain-hardeningexponents, Thompsen coefficients and other mechanical rock properties.

The Nature of Elastic Coefficients as they Pertain to Fractures

The mechanics of drilling a well provide a natural, in-situ, means tomeasure the deformation of a rock formation and gather data suitable fordetermining mechanical rock properties, because the penetration of thedrill bit is in and of itself accommodated by repeatedly fracturing therock formation by using the bit to generate forces on the rock formationthat are sufficient to overcome the failure strength of the rock,measurements of such in relation to the methods described here may beused in predictable ways to determine the presence of natural (in situ)fractures, fracture swarms (cluster of fractures), bedding planes, faultboundaries, and other information. In some instances, variations inmechanical rock properties are used to identify fractures, beddingplanes and the like.

Elastic coefficients that describe a relatively large deformation inresponse to the forces acting on a rock formation indicate the rockformation is weaker and less rigid. Therefore mapping the spatialvariations of the elastic coefficients provides information where thereare zones of weakness in the rock formation. If the nature andoccurrence of the zones of weakness in the rock formation as evidencedby changes in the elastic coefficients are localized in space orotherwise discrete relative to the surrounding elastic coefficients thatwould indicate the presence of a fracture or other mechanicaldiscontinuity. Systematic changes in the spatial distribution of theelastic coefficients as are derived by the method are used to identifymechanical discontinuities and geological boundaries of rock formationssuch as bedding planes or faults that act to separate or offset rockformation with different rock properties, where the differences in rockproperties are evidenced by the nature and distribution of the elasticcoefficients.

As will be understood from the present disclosure, mechanical strengthand deformation of the reservoir rock influences fracture creation,propagation and ability to maintain fracture permeability. FIG. 1 is asimplified diagram illustrating the difference between fractures inducedin a relatively brittle rock formation, and which may be includenaturally occurring fractures, versus fractures induced in a relativelyductile rock formation, which may include fewer or no natural fractures.As illustrated in FIG. 1A, a horizontal section 10 of a borehole hasbeen drilled through relatively brittle rock 12 and hydraulicallyfractured. In contrast, FIG. 1B illustrates a horizontal section 14 of aborehole drilled through relatively ductile rock 16 and hydraulicallyfractured. The fractures 18 created in the relatively brittle rock tendto penetrate deeper into the reservoir than the fracture 20 in ductilerock. Moreover, reservoir rock trending to the brittle end of the normalrange tends to have higher initial production rates and lower declinerates. The techniques described in this disclosure will provide newinformation for selecting hydrocarbon bearing zones by differentiatingbetween brittle rocks generally associated with larger fracture creationand better proppant support that is more permeable than ductile rockthat produces smaller, less productive fractures which are prone torapid compaction and closure and are less permeable. Similarly,techniques discussed herein may also identify areas where naturalfractures may exist provide similar advantages as brittle rock.Generally speaking, as will be understood from the disclosure, variousmechanical rock properties discussed herein will provide mechanismswhereby rock formations may be characterized, along the well bore, as tothe relative brittleness or ductileness, or the relative susceptibilityto stimulation techniques along the formation, which may include theidentification of existing fractures or at least rock propertiesindicative of existing fractures

Data Acquisition Techniques

FIG. 2 is a diagram of a bottom hole assembly portion 20 of a drillstring where the bottom hole assembly includes a drill bit 22, a mudmotor 24, a bit sub 26 including various measurement componentspositioned between the drill bit and the mud motor, and sections of pipe28 within a horizontal section 30 of a borehole, also referred to hereinas a well bore. The vibration data used in the described methodologiesmay be recorded as close to the source (drill bit) as practical to avoidattenuation through the bottom hole assembly. In one example, vibrationsare translated into data from acoustical signals interacting withaccelerometers. Similar vibrations data may be collected when drillingthrough concrete or some other known media, in the vertical boreholesection, as discussed in more detail below. One possible location forrecording is directly behind the drill bit and ahead of the mud motorusing the bit sub, although multiple bit subs may be used along thedrill string for geophysical processing of the desired signal. Drillinga wellbore involves using a portion of the weight of the drill string,known as weight on bit (WOB), to push the drill bit into a formation 30.The rotating force on the drill bit, known as torque on bit (TOB), cancome from the surface or from a mud motor close to the drill bit. Whenusing a mud motor, drilling mud is pumped down the drill string until itencounters the power drive section of the mud motor where a portion ofthe mud pressure and flow is converted into a rotational force, which ismechanically coupled to the bit to thereby place rotational torque onthe bit 22 to turn the bit. The rotational force on the bit can also beaugmented by or come exclusively from mechanisms at the surface on thedrilling rig.

The objective of the drilling process is to break the rock down intofragments that are small enough that they can be lifted and evacuatedfrom the wellbore with drilling fluids in order to continue toaccommodate the forwards motion of the bit. It should be noted that theaction of the drill bit on a rock formation causes the fracturing of therock formation, which fracturing is experienced as vibrations of thedrill, along the borehole to drill the hole, and in the formationimmediately adjacent the borehole. Moreover, the drill may encounterexisting fractures 34 while drilling. Hydraulic fracturing, in contrast,is a process that occurs during the completion phase by injecting fluidinto the borehole, typically with perforation clusters 22 in the casing,to initiate fractures 18/20 into the formation surrounding the borehole, as illustrated in FIG. 1.

In the illustrated diagram, the bit sub 26 is shown between the bit andthe mud motor. The bit sub is a cylindrical component that is operablycoupled between the mud motor 24 and the drill bit 22 in a way thatallows the mud motor to turn the bit. The bit sub includes a housing,typically in a cylindrical shape, or another mechanism to supportvarious possible measurement components 36 including strain gauges, oneor more accelerometers, pressure sensors, which may measure the pressureof the mud flow, temperature sensors which may measure the circulatingtemperature of the mud or other temperatures and which may be used toprovide correction or offset of measurements or calculations that varywith temperature, gyroscopes which may be used to measure inclinationand/or directional changes of the bit and string, and/or othercomponents to measure or derive the information discussed herein.

In one example, as shown in FIGS. 2 and 2A, the strain gauges aremounted on the bit sub to determine torque on the bit and the weight onthe bit (the force turning the bit and the force pushing the bit intothe rock formation). Various possible ways of mounting the straingauges, or combinations of strain gauges, are possible. Additionally, asshown in FIG. 2A, which is a representative front view of the bit 22,accelerometers are placed to measure axial, rotary, and/or lateralacceleration of the bit. Note, the bit axis is in the center of thecircle, whereas axial acceleration may be measured somewhat offset fromthe axis depending on the placement of the accelerometer. Accelerationmeasurement may be accomplished by using one or more multi-axisaccelerometers. The bit sub, or other such component, may also include aprocessor and memory to store computer executable instructions toimplement various possible methodologies, and possibly preprocess data,as well as a power source which may be one or more batteries. Datastorage, such as the memory or other data storage, is also provided tostore the collected data. The measurement components, alone or invarious possible combinations, may be provided in other locations of thedrill string in the general proximity of the drill bit.

Geomechanics in Relation to the Cutting Action of the Bit

FIGS. 3A-3C are a sequence of diagrams illustrating a close up view of acutter 32 portion of a drill bit in a borehole, slipping, sticking on aportion of rock, and then slipping loose when the forces on the bit aresufficient to overcome the rock causing the rock to fracture and the bitto rotate—collectively referred to as stick slip behavior. The rockdeformation mode for a PDC bit is shearing as opposed to a roller conebit which is punching. Models that describe drilling behavior are in alarge part informed on the mechanics of drilling with a roller cone bitand while these models have been extended for the application and use ofPDC bits, they suffer uniquely from their inherent inability toreconcile the fundamentally different nature of rock deformation. Aswill be appreciated from this disclosure, the innovative, new techniquesdisclosed here seek to advance the application and use of PDC bits, aswell as other bits, to characterize mechanical rock properties and inparticular for the identification of the nature and occurrences offractures. The figures describe the depth of cut in relationship to thearea and displacement of the fractures created in response to the forcesacting on the bit. The cutting action of a particular type of bit butshould not be construed to limit the method in the use of other types ofdrill bits that generate acoustic emissions from rock failure inresponse to the forces acting on the geometry and configuration of thebit.

More specifically, as the bit turns, the interaction of the bit with therock formation at any instant in time, produces a complex distributionof forces acting on the formation in connection with the bit anddrilling fluid system (e.g., the mud motor) where the orientation andmagnitudes of the forces acting on the rock formation are related to theconfiguration and geometry of the cutters on the bit. Generallyspeaking, drilling is not a smooth and consistent process. Instead,depending many things including the axial force on the bit, rotationaltorque on the bit, rock properties, and presence or absence or existingfractures, the bit cuts, gouges, spins, snags, and otherwise drills theborehole in a very complicated and varying fashion. In some instances,the complex distribution of forces acting on the formation isinsufficient to initially overcome the strength of the rock formation inrelation to the cutting action of the bit and the bit will stop rotatingor stick.

As illustrated in FIGS. 3A and 3B, as the cutter begins to stick, thetorque applied to the bit increases from a relatively steady value. Asthe forces, such as the illustrated torque on bit, applied to the bitchange either through manual or automated interaction with the surfacedrilling apparatus or through the non-linear feedback of elastic energystored within the drilling string or some combination of both, a newweight on the bit (WOB) and torque are delivered to the bit. Theseforces on one or more particular cutters will continue to load the rockelastically until such point (i) the rock begins to deform plasticallyand the deformation is concentrated along fracture planes and (ii) whenthose forces provides a sufficient distribution of forces to overcomethe failure strength of the rock formation, the bit will turn and rock,often snapping loose, and drilling will continue. As shown in FIG. 3C,when the forces overcome the rock, the torque will dramatically drop tothe relatively steady value, until the bit sticks again. Such a stickslip action may occur at varying frequencies and displacements and makehappen one or more times per revolution of the bit per cutter on thebit, thereby resulting in many such cutting behaviors each revolution ofthe bit.

Regardless of whether and the extent of stick slip behavior isexperienced, deformation and failure of the rock cause the bit tovibrate. During rock deformation and in particular when the bitovercomes the rock strength at failure, stored elastic strain energy isreleased in the form of acoustic emissions. In some instances, the bitis fracturing rock and may intersect fractures and existing mechanicaldiscontinuities. In some instances, the bit may reactivate existingfractures, which may itself generate a distinct acoustical signal in theform of an induced bit vibration.

On the Processing of Drilling Vibrations

As introduced above, drill bits 22 typically include many cutters 32arranged with a geometry and configuration designed to generatesufficient forces to overcome the failure strength of the rock formation30 based on the nature of the rock formation expected to be encounteredwhen drilling a well. During a single rotation of the bit, at least one,but typically many of the cutters will overcome the failure strength ofthe rock formation and produce a plethora of acoustical emissionsrelated to the scraping, cutting, fracturing, and other interactionsbetween the bit and the rock formation. The tool may include variouspossible mechanisms, including a reed switch or gyroscopes, whichmeasure revolutions per minute and provide information of each rotationof the bit. There are as many as 30 cutter heads, so each rotation maycause hundreds of acoustic pulses.

Because of the stochastic nature of acoustic emissions or in relation tothe nearly simultaneous initiation and propagation of multiple fracturesat the cutting face, the implementation of the method discussed hereinmay use statistical methods and signal analysis tools. In one possiblemethodology, the RMS measurement technique determines the energy of thesignal despite the shape of its waveform. This is important because themany simultaneous events will create complex waveforms with constructiveand destructive interferences.

Given the stochastic nature of the acoustic emissions generated by thecutting action of the bit, it is expected that the more times the bitturns per unit time, the higher the rate of acoustic emissions. RMSlevels obtained for a time window that underwent two revolutions of thebit would be expected to have higher RMS levels than those obtainedduring one revolution of the bit in the same time window, all otherthings being equal.

Measurements while drilling show that the bit speed can vary wildly anderratically during drilling operations. In some instances the bit cancompletely stick and then slip again as more force is gradually applied.If the stick-slip behavior of the drill bit is not accounted for in thegeophysical signal processing, then the variations in the fracturemeasurements may be confused by with variations in the bit speed and notvariations in the fracturing behavior of the rock. When the time windowused to measure the RMS level of the signals that have been extractedfrom the drilling vibrations is normalized with respect to the bitspeed, where the bit speed is recording using a gyro to sample thechanges in position of the bit with respect to time or a magnet which isused to inform the position of the bit with respect to time, thisnormalized measurement is understood to provide a level of theacoustical emission activity generated by the fracturing of a rockformation in relation to the cutting action of the bit. The time windowwhere data is gathered may be tied directly to bit rotation by having atime window set based on counts of the bit revolution. In otherinstances, the time window may be set, and the energy (from vibrations)may be normalized to account for some set turns of the bit, such as oneturn of the bit.

The implementation of the technique as described uses signal processingtechniques, such as Fourier transforms, bandpass filtering or otherfiltering, or combinations thereof, to calculate the motion of the bitand the forces on the bit from the amplitudes and frequencies of theacoustical signals recorded by the MWD apparatus (e.g. bit sub 22) thatare generated in response to the cutting action of the bit (e.g., thedrilling vibrations propagate up the drill string as acoustic wavessometimes referred to as collar waves or tool mode, where they arerecorded as acoustical signals by the MWD apparatus). Hence,accelerometers capturing acceleration data can be used to obtaininformation of the forces on the bit. Forces on the bit may also bemeasured using strain gauges, in one possible implementation. The motionof the bit, which may be recognized as displacements, and the forces onthe bit are used to populate a stress/strain relationship that allowsthe computation of mechanical rock properties. Mechanical rockproperties may be analyzed relative to a baseline (such as an averageover some wellbore distance) to identify locations where the mechanicalproperties of a rock formation change as the bit encounters a mechanicaldiscontinuity or other geological discontinuity all other things beingequal. In some instances, rock properties may be used to identify suchlocations through computations based on assumptions of the rockformation, and comparisons thereof, or otherwise.

Referring to FIG. 4, to account for the effects of the drillingefficiencies and the possibility of stick-slip behavior while conductingdrilling operations, in one specific implementation, the method may usemeasurements of the drilling efficiencies, such as revolutions perminute (RPM) to normalize the geophysical signal processing of thedrilling vibrations in order to compare the results of the signalprocessing along the length of the borehole in accordance with themethods provided. It is understood through the normalization that thevariations in the signal levels, such as the RMS levels, are nowcorrected in account of the changes in drilling efficiencies along thetrajectory of the borehole.

As such, the RMS levels, obtained as shown above, for example, arerelated to changes in the mechanical rock properties while the drillintersects areas of differing properties. Stated differently, it may behelpful to compensate or normalize for changes in the rotational speedof the bit (RPM) by using a time window commensurate with the rotationof the bit. Such normalization may also be useful to correct thegeophysical signal processing in accordance with drilling operationswhere the drill string is rotated from surface in conjunction with themud motor turning the bit, as opposed to situations where the mud motoris operating but the string is sliding not being rotated from thesurface.

In a further elaboration of the method, the signal processing techniquessystematically relate measurements of the forces acting on a rockformation in connection with the drill bit and drilling fluid system(stress) to the variations in the fracturing of a rock formation inresponse to the cutting action of the bit (strain) to obtain innovative,new stress-strain relationships where the application and use of thestress-strain relationships allow for the derivation of elasticcoefficients for the stress-strain relationships. Relative variations ineither one or a combination of the elastic coefficients may be used toidentify the nature and occurrence of fractures, fracture swarms andother mechanical discontinuities and geological boundaries such asbedding planes and/or faults that offset or otherwise separate rockformations with different mechanical rock properties.

Accounting for the drilling efficiencies is also a consideration in theimplementation of the approach, because the wear on the drill bit as therock formation is drilled will change the configuration and geometry ofthe cutters on the drill bit. The mechanical wear of the drill bit willaffect the distribution of the forces acting on the formation inconnection with the drill bit and as the well is progressed, the manualor automated application of forces on the drilling apparatus change toaccount for the wear and tear of the bit.

The application and use of the stress-strain technique according themethod employed herein is understood to normalize the effects of reduceddrilling efficiencies caused by bit wear on the derivation of themechanical rock properties because the forces used in the stress-strainrelationship are obtained from the forces acting on the bit, where theforces needed to overcome the strength of the rock are increased inrelation to the penetration or depth of cut of the worn bit tooth intothe formation.

The acoustical emissions associated with the deformation and failure ofthe rock formation while drilling are generally too minute and/or tooattenuated by the intervening rock to be detectable at the surface(which may be hundreds or thousands of feet above the borehole). Becauseof the amount of energy released is generally expected to be slight andof relatively high frequency, the radiated waves are best viewed whentransmitted from the cutting face through the bit and bottom holeassembly where they propagate along the drill string throughacoustically conductive steel as a direct tool arrival and contribute tothe vibration of the drill string. The drilling vibrations can berecorded on instrumentation that is sensitive to their nature andpresence. Stated differently, one aspect of the present disclosureinvolves a drilling tool assembly including sensors and processingelectronics (e.g., the accelerometers and/or strain gauges in the bitsub 26 proximate the bit 22) that are positioned to detect and recordthe radiated waves from the drilling induced fracturing, which mayfurther involve identification and/or characterization of existingmechanical discontinuities, such as fractures or geological boundaries,such as faults or bedding planes.

A Specific Data Logging Technique

In one specific implementation, a form of measurement while drilling(MWD) system or tool is employed. The MWD system uses sensors designedto measures vibration. The MWD system may also measure forces on thebit, and other parameters such as bit speed, which may be expressed asrevolutions per minute, the fluid pressures, and temperature of thedrilling mud or environment proximate the bit sub. The system may alsoinclude gyroscopes to obtain the orientation of the cutting face of thedrill bit, in some implementations. In one specific embodiment, the MWDtool includes at least one receiver, which may include accelerometersmounted on or proximate the bottom hole assembly to record the drillingvibrations and associated acoustic emissions. In some implementations,the MWD may further include electrical, mechanical, and/or otherfiltering mechanisms to processes the data to remove unwanted noise orto record the data without unwanted noise. In certain instances, stagesof filtering may be applied both prior to recording, and after recordingbut prior to processing, to remove unwanted data, or as much asnecessary or possible. In an alternative enablement, the signals may betransmitted to the surface for storage and processing. In someapplications it may be desirable to process the acoustical signals, suchas through the processor, on board the logging tool for transmission ofthe significantly data-reduced processed signal to surface in real time.

Once the drilling dynamics data is collected and processed, the resultsare correlated back to the measured depth of the well using precisemeasurements of the length of drill string components as they arelowered into the well. Gamma ray LWD and casing collar measurements canfurther be used to correlate the absolute location of data processedfrom the BHA collection point or points to determine a more reliablelocation of the bit in relation to the subsurface.

Noise Attenuation Techniques

Because of the amount of energy released is generally expected to beslight and of relatively high frequency, the radiated waves are bestviewed when transmitted from the cutting face (with cutters 32) to thebit and bottom hole assembly, where they may further propagate and areknown as the direct tool arrival or collar wave and contribute to thevibration of the drill string. The acoustic emissions may be measured byaccelerometers, transducers, or other devices sensitive to particlemotion.

Drilling induced vibrations that are generated by the interaction of thebit with the rock formation will have harmonic frequencies that arerelated to the rotational speed of the bit. Most of the harmonicvibrations are expected to be low frequency. The amplitudes of theacoustical emissions in the frequency ranges of the harmonic frequenciesare usually much less than the amplitude of the harmonic drillingvibrations. In one implementation, the harmonic drilling vibrations areremoved by a filter, such as a high pass filter or bandpass filter,which may be implemented in the bit sub processor, or may be applied tothe stored data for later processing after download of the data from thebit sub memory, that passes signal frequencies that are higher thanfrequencies related to the harmonic drilling vibrations, and which mayalso eliminate frequencies above those possibly related to fracturecharacteristics. Other filter types and frequency characteristics willbe possible depending on various factors including, but not limited to,the rotational speed of the bit, the type of bit, the rockcharacteristics, the positioning of sensors, mud motor characteristics,and other attributes.

Another consideration with respect to the variations in the amplitudeand frequency of the acoustic emissions is the interference of thedirect tool arrival by the generation and transmission of other wavemodes that are also excited by the drilling operations including thewave modes excited by the release of energy from the fracturing of therock formation.

In addition to the acoustical emissions created by the fracturing of arock formation in response to the cutting action of the bit, thedrilling vibrations that are being recorded will also generate wavemodes that propagate though the formation surrounding the wellbore andthe fluids within the wellbore that can interfere with the acousticalemissions that are related to the fracturing of a rock formation at thecutting face and bias the measurements of the mechanical rockproperties. For some frequencies, the other wave modes will have higheramplitudes than the amplitudes of the acoustical emissions at thatparticular frequency. These other wave modes will result from thepropagation of various guided waves in the drilling fluid between thebottom hole assembly and the wellbore such as Stonely waves, tube wavesand direct fluid waves such as the fluid compressional waves. Other wavemodes that can interfere with the direct tool arrival are surface wavesthat propagate and refract energy along the interface between the wellbore and the fluid such as compressional head waves, surfacecompressional waves and shear body waves. These waves all have thepotential to interfere with the propagation of energy from the directtool arrival and if their presence is included in the signal processingcould bias the calculations related to the amplitudes and frequenciesused to describe the nature and occurrence of the fracturing.

To decrease the possibility of interference with the other wave modes,in one embodiment the sensor is mounted internally using in a plug whichwill effectively isolate the sensor from the wave modes propagatingthrough the formation and through the fluids. For example, in the caseof accelerometers, the accelerometers are mounted within the bit sub (orother component). External waves that are carried by the formation andfluid in the annulus 36 will therefore be mechanically isolated by theinternal position of the sensor relative to the other waves. Aninternally mounted sensor will be respond mainly to the vibrationsrelated to direct tool arrival (the vibrations caused by the interactionof the bit with the formation).

Because the steel used in the construction of a bottom assembly has asignificantly higher quality factor, Q of 10,000, that rock formationsand fluids where Q ranges are typically from 1 to 100, the waves thatpropagate along the collar known as the collar wave or tool arrival willexperience much less attenuation than signals recorded by the sensorthat have traveled through the formation or the fluid media. Because ofthe low attenuation of the waves propagating along the steel drillcollar relative to the waves propagating through the formations anddrilling fluids, it could also be possible to naturally attenuate thevarious unwanted modes of propagation by placing the receiver at adistance that is far enough away from the bit to attenuate the otherunwanted formation and fluid wave modes that will interfere with thedirect tool arrival, but not so far as to lose the valuable highfrequency information carried by the collar wave or tool wave that isneeded to be recorded to calculate the size and displacement of thefracturing. The distances needed to attenuate the unwanted modes can bedetermined by expressions that relate the energy loss per cycle duringtransmission for a given quality factor. In this embodiment the locationof the bottom hole assembly in relation to the location of the bit isplaced at a distance behind the bit to achieve this attenuation.Alternatively or additionally, to avoid the interference of fluid andformation arrivals with the collar arrival the receiver may be coupledto the drill string at the surface of the well where the drill stringhas yet to enter the subsurface and borehole that contains the drillingfluids.

In another embodiment, a plurality of receivers are arranged along thestring behind the drill bit to record the acoustical signals generatedby the release of elastic energy at the cutting face. In one example,the receivers are spaced within centimeters or millimeters and may beplaced in array mounted on a portion of the bit sub. The spacing of thereceivers may form an array, with the spacing determined by thefrequency range of the acoustical signals and the velocity ofpropagation in the steel or other material. Because the velocity ofpropagation in the steel is typically much faster than most formationand fluid velocities which control the nature of propagation of theinterfering modes and is known with a high degree of certainty, spatialfilters such as FK filters that pass signals that propagate atvelocities that are consistent with wave transmission through steel andattenuate events with slower velocities. These filters can be used toseparate the direct tool arrival waves from the other interferingwaveforms. When using an array of receivers the signal-to-noise ratiocan be further increased by using geophysical signal processingtechniques to filter the data by stacking the signals over the array.The nature of the stacking depends on the configuration of the array ofreceivers and whether the receivers are deployed in a linear, bipolar orradial array.

In some instances the stacking can be used to isolate various modes thatare propagated by the tool arrival, such as the compressional wave,transverse wave or quadrupole wave. These other tool modes can also beprocessed using geophysical signal processing techniques to determinethe fracturing of a rock formation in relation to the cutting action ofthe bit.

The quadrupole is a direct tool mode that does not propagate direct toolarrivals above a cutoff frequency, where the cutoff frequency is relatedto the diameter and thickness of the steel. The amplitudes andfrequencies of these other modes provides useful though otherwise bandlimited information that may be used in relation to the geophysicalsignal processing techniques in order to specify how the size anddisplacement of the fractures would be responsible for generating theseother wave modes.

The analysis of the signals extracted from the drilling vibrationsshould not be limited to the case of the amplitudes and frequencies ofthe direct tool arrival. In another embodiment, the receiver array canbe used to reject the direct tool arrival or collar wave and pass otherarrivals related to other modes of transmission, where the modes oftransmission may be through the rock formation or drilling fluid systembased on their velocity of propagation and frequency content through themedia. These other modes of propagation may be used in preference to thedirect tool arrival when the other modes of wave propagation containsignals related to the fracturing of the formation in relation to thecutting action of the bit that are of interest to the variations of theRMS levels of the signals as described by the method.

The Application and Use of Microseismic Signal Processing Techniques inRelation to the Method

In a further elaboration of the method, drilling vibrations generated bythe cutting action of the bit may be processed using geophysical signalprocessing techniques that are conventionally recognized as appropriatefor the analysis of microearthquake source mechanisms. In one example,the depth of cut or penetration per revolution of the bit is obtained byusing signal processing techniques to measure the sizes anddisplacements of fractures that result from the deformation and failureof the rock formation in relation to the cutting action of the bit, toestimate the zero-frequency level (ZFL) of the displacement spectra. TheZFL of the displacement is the static offset level here understood torepresent the penetration of the bit. Under this consideration and inrelation to the technique it provides high resolution motion of the bitthat is taken to be the depth of cut or penetration of the bit for eachrevolution of the bit.

FIG. 5 illustrates a displacement spectra for one revolution of a bit.The graph of displacement spectra has a y-axis of rotary displacementamplitude at a range of frequencies (x-axis) over which amplitudemeasurements are taken (based on the sampling frequency.) Thedisplacement spectra is obtained by integrating the Fourier transform ofthe time domain measurement of the acceleration data (bit vibrationdata) twice in the frequency domain, or the angular velocity time seriesonce where it is understood that the accelerations may be axial,lateral, rotary or a combination of those channels. Rotary displacementmay be in the form of radians. The ZFL (zero frequency level) determinedby the method is where the displacement spectra theoretically intersectsthe zero-frequency axis. The ZFL is directly proportional to the averagedisplacements of the fractures initiated by the bit, and to thedisplacement of the bit per revolution.

Generally speaking, because high resolution displacement data isachievable, the system can detect relative changes of displacement perrevolution of the bit or displacement per time or rate of penetration(ROP) and thereby determine when a fracture is encountered as thedisplacement will be greater relative to areas where fractures are notencountered. With respect to the method, the increase in displacementwould indicate a change in the hardness of the formation, where hardnessincreases with displacement all other things being equal.

Observations of strong ground motion generated by earthquakes suggestthat the time series recordings of the P- and S-wave signals can bereasonably and effectively treated as band limited white noise where theamplitudes and frequencies of the signals are controlled by theinteraction of many smaller faults and fracture patches rupturingsimultaneously and the band limitations in the absence of attenuationare related to the displacements and rupture dimensions of earthquakesource. Therefore, observations involving the simultaneous occurrence ofmultiple acoustic emissions generated by fracturing at the face of thebit and then transmitted through steel where the signals are expected toundergo little attenuation indicates that the application and use ofmodels that are generally used to describe the source mechanisms of anearthquake can be used to describe the aggregate sizes and displacementsof many fractures being generated simultaneously from the repeatedturning of the many cutters on the drill bit when overcoming the rockstrength to accommodate the motion of the bit.

The estimation of the size and displacement of the fracturing generatedby the cutting action or motion of the bit follows directly from thedisplay of the data that is present in FIG. 5. Here the model forrepresenting the microearthquake source mechanism that estimate the sizeand displacement of the fracturing typically involves the applicationand use of a two parameter spectral model where here the two parametersused to describe the model are the zero-frequency level of thedisplacement spectra and the corner frequency. It should be appreciatedthat other source parameter models such as the RMS stress drop that alsouse the amplitude and frequencies of the signals to estimate the sizeand displacement of fracture and can also be considered and the use of atwo parameter spectral model should not limit the scope of this method.

In the two parameter model of the microearthquake source mechanism, thedisplacement of the bit as determined by the displacements on thefractures is related to the zero frequency level of the displacementspectra. The zero frequency level of the displacement spectra is thesame as the static offset. The low-frequency content may be modified bythe receivers and electronics used to record the signals on the MWDassembly and the possible use of band pass filters in relation to thesignal processing to eliminate the low frequency drilling harmonics. Thetechnique has advantages because it (i) provides an estimate of the ZFLusing band-limited data that may not always be reliable in the lower endof the amplitude spectra, (ii) provides a reliable technique toobjectively select the ZFL without manual or visual biases, (iii) can beautomated in a computer implemented fashion to handle the large volumesof data typically collected by an MWD apparatus as used to collect datain relation to the method.

Geophysical signal processing techniques that use relationships betweenthe power spectral density of the displacement and velocity spectrum areused to calculate this value (ZFL) based on the functional mathematicalrepresentation of the earthquake source spectra as described by the twoparameter spectral model of the earthquake source provide an objectivetechnique to overcome the expected poor signal to noise ratios of thelow frequency acoustical information.

Thus, ZFL is understood to represent the displacement or penetration ofthe bit. In one specific example, the ZFL is a measurement of thedisplacement (e.g., in millimeters or inches or angular displacementssuch as a radians) of the bit per turn of the bit. If the displacementis the axial displacement per revolution of the bit, then thisdisplacement can be used to inform a depth of cut in terms ofdisplacement per revolution. If time taken to make the one turn of thebit is used to describe the axial displacement or penetration of thebit, then it is taken that this may be used to determine rate ofpenetration. Typical depths of cut as estimated by the method range from0.01 inches per revolution to 0.1 inches per revolution

The size or the radius of the fractures can also be determined by thefrequency content of the acoustical signal. Signals with higherfrequency content generally correspond to smaller fracture areas. Thischaracteristic of the frequency spectra that is used to determine thesize of the event is typically referred to as the corner frequency (FIG.5). There is a linear, relationship between the corner frequency of thedisplacement spectra and the size of the event. Because of attenuationof the signal during the transmission, estimates of the corner frequencymay be compromised if the sensor is placed too far from the bit. Thus,having the sensors as close to the bit as possible, or at least not sofar that attenuation is significant, may be a consideration for someimplementations discussed herein. Limitations in the sensor and therecording electronics and the filters employed to extract the signalsmay also limit the useable bandwidth.

Geophysical signal processing techniques that use relationships betweenthe power spectral density of the displacement and power spectraldensity of the velocity based on the functional mathematicalrepresentation of the earthquake source spectra as is described by thetwo parameter model of the earthquake source that can be used tocalculate the corner frequency. A relatively high corner frequency mayrepresent unfractured rock whereas relatively lower corner frequency mayindicate the presence of a fracture.

When the bandwidth is limited, the minimum fracture size detected willbe set to a threshold based on a cutoff frequency. Any fracturing of arock formation below this threshold will not contribute to thedetermination of the fracturing of the rock formation and thedeformation of the rock formation. Thus, in the application of themethod, some rock formations may undergo fracturing in response to thecutting action of the bit where the fracturing is not detected by themethod employed here, either because the frequencies of which tofracture energy occur are too high or the signal to noise ratio is toolow. The occurrence of these scenarios would suggest that the bit is noteffectively penetrating the formation and that the rock propertiesderived in this instance would represent the limiting rock propertiesand limit the specification of the deformation of the rock in responseto forces generated by the emplacement of hydraulic fractures where anysuch deformation in relation to the cutting action of the bit would betaken as insufficient to do so.

In order to make meaningful comparisons of the fracture sizes anddisplacements along the well bore, the time window that is used toprocess the signals extracted from the drilling vibrations is based onthe bit speed, in one specific implementation. One way to normalize thetime window relative to the bit speed is to specify the time windowaccording to the time needed to make one revolution of the bit. If thebit speed were 120 RPM then the time window would be 500 ms, while ifthe bit speed were 60 RPM then the time window would need to be 1000 msto obtain an equivalent measurement of the fracturing generated inresponse to the cutting action of the bit. Another way to normalize thetime window to account for variations in the bit speed along thewellbore would be to normalize the temporal frequency of the spectraldensity by converting from cycles per second to cycles per revolution.At a bit speed of 120 RPM, the bit would make two revolutions in asecond and therefore at cycle per second would be normalized to twocycles per revolution, while a bit speed of 60 RPM would make onerevolution in a second therefore one cycle per second would benormalized to one cycle per revolution. In one embodiment, the BHA isinstrumented to measure the bit speed at sufficient resolution tospecify either a time window equivalent to one rotation of the bit ornormalize the spectral density by converting the temporal frequency tocycles per revolution.

As such, the spatial variations in the measurements as obtained througha combination of one or more of the measurements such as the RMSacceleration or the sizes and displacements of the fractures areunderstood to correspond to the nature and occurrence of deformation andfailure in relation to the cutting action of the bit and as such aretaken to represent the spatial variations in mechanical rock properties.So, an increasing value of ZFL, relative to a baseline, represents theintersection of the bit with zone of mechanically weaker rock that maybe determined to be a fracture, swarm of fractures (e.g., fracture 34)in accordance with for example a stress-strain relationship. Therefore,the geophysical signal processing techniques employed by the method mayinvolve statistical descriptions such as measuring the RMS level of theacoustical emissions or the application and use of geophysical signalprocessing techniques that are generally recognized as appropriate forthe analysis of microearthquake source mechanisms to describe thedeformation and failure of a rock formation in relation to the cuttingaction of the bit, where the spatial variations in the measurements(e.g., changes in ZFL or corner frequencies) relative to some average orbaseline level are used to identify mechanical discontinuities orgeological formations as they are encountered or crossed by the bit.

In another embodiment, referring now to FIG. 6, the geophysical signalprocessing methods may obtain measurements related to the instantaneouschange in the inclination of the bit relative to the average directionof the bit to describe a mechanical discontinuity or geologicalboundary. Instantaneous change in inclination of the drill bit relativeto a long-term average inclination can be obtained using a sensor or anarray of sensors configured to record and extract acoustical signals inrelation to the independent spatial axes of the drilling vibrations. Themagnitude of the deflection of the inclination of the drill bit relativeto a long-term trend in the direction of the drill bit depends on theorientation of the mechanical discontinuity or geological boundary withrespect to the cutting face of the drill bit and therefore thedeflection is understood to indicate a change in the mechanical rockproperties. So, for example, with a multi-axis accelerometer measuringaxial acceleration and lateral or rotary acceleration of the bit, theratio of the axial and lateral or rotary displacements may be treated asan inclination, as illustrated in the respective inclination logs 60(prior to the fracture, while the bit 22 intersects the existingfracture 34). The lateral or rotary acceleration will increaserelatively when the bit is deflected from axial movement, such as whenthe bit encounters an angled discontinuity transverse the borehole.

The Concept of Specific Energy in Relation to the Method

In addition to acoustical information that informs the displacement ofthe bit, aspects of the present disclosure may further involve forceinformation. While conducting drilling operations the energy defined asthe energy needed to remove a volume of rock is useful to describe theefficiency of the drilling operation. The specific energy is a term thatdescribes the minimum amount of work needed to remove a certain volumeof rock. Descriptions of the specific energy are useful to understandthe variations in rate of penetration or the depth of cut in relation tothe forces acting on the bit. Unlike conventional methods, the specificenergy here may be calculated from the aforementioned acousticalprocessing techniques in and of themselves or in conjunction withmeasurements of torque on bit and/or force on bit.

The size and displacement of the fractures that form in response to thecutting action of the bit controls depth of cut into a rock formationand subsequently the rate of penetration of the drill bit through a rockformation. In a method described here, the work done per the volume ofrock removed is determined by the displacement of the fractures asmultiplied by the area of the bit. The work is computed from the forcesacting on the bit multiplied by the displacement of the bit. Asdiscussed above, the displacements of the bit are obtained through theanalysis of the drilling vibration using geophysical signal processingtechniques that are appropriate for the analysis of microearthquakesource mechanisms as is provided by the above discussed method.

There are two components of the specific energy: one that is normal tothe bit and another that is tangential to the bit or rotary energy.Because the rotary specific energy is proportionally related to thetorque per unit of displacement, it provides a measure of the rockstrength or the minimum energy needed to drill and the application anduse of the rotary specific energy in this regard takes the form of astress-strain relationship (FIG. 8 and others).

In the method provided here, the displacement of the bit as it relatesto the fracturing of the formation is evidenced through the RMS level ofthe measurements or the displacement of the bit is evidenced by thedisplacement on the fractures as provided through the microearthquakesource parameter measurements and the volume of rock removed isproportional to the fracture areas and the fracture displacement areaveraged over all of the fractures provides in the period of timeprocessed for the time period that is analyzed for example in a singleturn of the bit. In one embodiment of the method, the volume of rockexcavated is specified by the product of the aggregate area of thefractures and the average displacement of the fractures removed by oneturn of the bit.

For a bit that is turning at a rate of 120 RPM this would involve usinga time window of 500 ms to the average area and average the displacementfor one revolution of the drill bit. It should be appreciated that ininstances when the rock properties are varying slowly or the MWDmeasurements are updated at rates less than the time period of one bitrevolution the method is not limited to periods that are specified bythe turning rate of the bit.

The rotational specific energy is the torque over the average fracturedisplacement per revolution as obtained by the analysis of the vibrationdata disclosed by this technique, provides a novel, innovativestress-strain relationship (FIG. 8 and others). For a given rock type,this is expected to be a linear relationship where the slope of the lineis related to an elastic coefficient describing the strength of the rock(e.g., specified rock strength 1 (70), specified rock strength 2 (72),and specified rock strength 3 (74)). Because the measurements areobtained while drilling, the strength of the rock actually determined byand is intrinsically related to the fracturing in response to thecutting action of the bit. When using a polycrystalline diamond compact(PDC) bit, for example, the strength of the rock determined from thisstress strain relationship would be directly related to the shearstrength of the rock. Stress-strain measurements obtained using themethod disclosed can be used to characterize the elastic coefficients ofthe rock formation.

Innovative Stress Strain Relationships as are Provided by the Method

The stress strain relationships employed by the method are populatedfrom measurements taken while drilling. In one instance, the strain isunderstood to be related to the depth of cut or the penetration perrevolution of the bit which is determined by differencing the spatiallocation at two instances in time versus the number of revolutions takenfor the bit to travel that distance. In accordance with the methods setout here in, where (i) the drilling vibrations are understood torepresent the deformation and failure of a rock formation in response tocutting action of the bit in order to accommodate the forward motion ofthe bit through a rock formation, and (ii) the drilling vibrations asprocessed through the signal processing techniques to evidence themotion of the bit per turn of the bit are understood to represent strainand (iii) the RMS acceleration as obtained through the geophysicalsignal processing techniques to evidence the forces acting on the bitper turn of the bit are used to populate the variable in relation to astress-strain constitutive equation. In a further elaboration, thesestrain measurements can be related to the forces such as the weight onbit and torque on bit acting on the formation in connection with the bitand drilling fluid system to provide a diagram of a generalstress-strain relationship. By relating the orientation and magnitude ofthe stress with respect to the orientation and magnitudes of the strain,where the orientations and magnitudes of the stress and strain arerelated to the geographical coordinates of the well, multiplestress-strain relationships can be established to determine the elasticcoefficients of a rock formation. Where the well is drill perpendicularto the maximum horizontal compressive stress, it is understood thatthese stress-strain relationships are expressed in the principle axes.

In general, the deformation of a homogeneous isotropic rock formationcan be specified by two elastic coefficients. As the complexity of therock formation increases through the presence of mechanicaldiscontinuities and geological boundaries, the number of elasticcoefficients needed to fully describe the deformation of a rockformation in response to the forces acting on the formation in generalincreases.

The simplest elastic coefficient would be to relate the WOB to thestrain generated by the cutting action of the bit where the strain isthe displacement in relation to a length made by one turn of the bitwhere the time windows used for the geophysical signal processingtechniques are related to the bit speed. In one preferred embodiment ofthe method, the WOB would be obtained from the RMS acceleration, wherethe component of acceleration is oriented parallel to the borehole andas is illustrated.

For a transverse isotropic (TI) media, the stress strain relationshipconstitutive equation is illustrated in FIG. 8A (using MWD data) withattendant PR intercept and YME slope information illustrated in thecurves of FIG. 8B.

Where:

-   -   E is Young's Modulus of Elasticity (YME),    -   ν is Poisson's ratio (PR)    -   σ1=TOB    -   σ3=WOB    -   ε3=Axial displacement spectra ZFL

This stress-strain relationship would in general be proportional to aYoung's modulus of elasticity where YME is determined parallel to thedirection of drilling. The method allows for the determination ofelastic coefficients for transverse isotropic elastic media, where theassumption of transverse isotropic elasticity is understood to bereasonable approximation to describe the rock deformation by obtainingthe stress-strain relationship to the orientation geological boundarieswith respect to the inclination of the bit of the well being drilled.This could be accomplished using FIG. 8 or FIG. 100 where the elasticcoefficients as described are proportional to Young's Modulus and thePoisson's ratio of a rock formation.

In another implementation more particularly shown in FIGS. 9A and 9B,the rock strength is specified by the stress-strain relationships basedon the orientation of the well in relation to the bedding planes of thehydrocarbon bearing formation and the orientation of the well withrespect to the principle axes of tectonic stress or the state-of-stressacting on the rock formation. This method allows for the determinationof elastic coefficients for a transverse isotropic (TI) elastic media,where the assumption of transverse elasticity is taken as a reasonableapproximation to describe the rock deformation by obtaining thestress-strain relationship to the orientation geological boundaries withrespect to the inclination of the bit of the well being drilled. Thiscould be accomplished using the above description where the elasticcoefficients as described are proportional to Young's Modulus and thePoisson's ratio of a rock formation.

The loading conditions on a rock formation are given by the forcesacting on the rock formation in connection with the drilling apparatus(e.g., weigh on bit and/or torque on bit) and drilling fluid system(e.g., annular pressure) and the deformation of the rock formation aredescribed by the displacements of the bit (e.g., axial and lateral orrotary displacements as measured by accelerometers in the bit sub). Theconstitutive equations are described with respect to transverseisotropic media. As shown in FIG. 9A-9B, transverse isotropic mediainvolves a layering of media (sometimes referred to as “layercake”).Transverse isotropic media involves a layering of media that is normalto a plane of isotropy—meaning the media is relatively uniform aroundthe axis of symmetry. FIG. 9A illustrates a case where the axis 90 ofmaterial symmetry is parallel to the borehole 94 (and parallel to thebit axis 96 drilling the borehole). FIG. 9B illustrates a case where theaxis of material symmetry 98 is perpendicular to the borehole. In onepossible implementation, it is assumed that the media is eithervertically transverse isotropic (VTI) or it is horizontally transverseisotropic (HTI). VTI is a case where the axis of symmetry is verticallyoriented (layers are horizontal with respect to the free surface). HTIis a case where the axis of symmetry is horizontally oriented (where theanisotropy is understood to involve a layering of the formation that isvertical with respect to the free surface and further where it isunderstood that fractures represent a case of vertical layering withrespect to the free surface). Thus, in the case of a vertical well, FIG.9A illustrates vertical transverse isotropy (the vertical borehole isparallel to a vertical axis of isotropic layer symmetry) and alsoillustrates, in the case of a horizontal well, horizontal transverseisotropy (the horizontal borehole is parallel to a horizontal axis ofisotropic layer symmetry). In contrast, in the case of a horizontalwell, FIG. 9B illustrates vertical transverse isotropy (the horizontalborehole is perpendicular a vertical axis of isotropic layer symmetry)and also illustrates, in the case of a vertical well, horizontaltransverse isotropy (the vertical borehole is parallel a horizontal axisof isotropic layer symmetry).

FIGS. 9A and 9B also illustrate the variables for the constitutivestress-strain equation (FIG. 8 and FIG. 100) of a transverse isotropicmedia. The stress strain variables are populated based on either (i)force measurements (e.g., from strain gauges) and/or (ii) theacceleration measurements (e.g., from accelerometers) or populated onlyform acceleration data. More specifically, in the first case, thestress-strain variables are populated with data related to WOB, TOB oraxial or lateral or rotary displacements. In the second case, the stressstrain variables the forces are populated with axial and lateral orrotary acceleration data, and the strain from the axial or lateral orrotary displacements (which may be available by integrating the axial orlateral or rotary acceleration spectra twice in the frequency domain).Generally speaking, Poisson's ratio (and/or Young's modulus) is computedfrom stress-strain constitutive equations populated from measurementstaken while drilling along a borehole under the assumption that thetransverse axis of material symmetry is parallel to the borehole (FIG.9A) and under the assumption that the transverse axis of materialsymmetry is perpendicular to the borehole (FIG. 9B).

When the rock formation (media) is isotropic, the two equations willgenerate values of PR or YME that generally track each other, meaningthat under either assumptive case, the drill bit will react similarly asit drills through relatively uniform rock and therefore the twocalculations of PR and YME, while different, will nonetheless track eachother. If, however, the borehole intersects discontinuities and or morethe media is anisotropic, the computations of PR and/or YME will nolonger track each other. The stress-strain relationships may be recastinto other equivalent forms, and the particular arrangement shown inFIGS. 8, 9, and 10 are taken for the sake of conveniences and should notbe considered limiting. In a particularly useful manner as shown in FIG.11, the constitutive equations could be re-arranged so that the Young'smodulus term is determined using the slope of a linear relationship ifthe equations were cast in terms of the ratio of the forces actingparallel to the axis of symmetry to the forces acting perpendicular tothe axis of material symmetry.

In certain instances, such as drilling under high confining annularpressure, the confining pressure may be more accurately described usingthe annular pressure instead of WOB. In such a case, AP would be used inplace of WOB for the cases illustrated in FIGS. 9A and 9B, and possiblyothers.

Some of the stress strain relationships are set out in terms of bi-axialloading accounting for weight on bit and torque on bit. However, itshould be understood that an implementation accounting for tri-axialloading may nonetheless use and account for WOB and TOB or the RMSacceleration.

Thus, in the application of the stress-strain relationships for alayered media, where the bedding planes are horizontal a vertical welland a horizontal well are drilled through the same rock formation where:

-   -   1. Separate stress-strain relationships that describe the        elastic coefficients when the bit is cutting perpendicular to        the geological boundaries such as the bedding planes and when        the bit is cutting parallel to the geological boundaries would        enable at least four elastic coefficients to be determined which        can be used to describe in general, stress-strain relationships        of transverse isotropic rock formations.    -   2. Separate stress-strain relationships that describe the        elastic coefficients when the bit is cutting parallel to the        direction of maximum horizontal compressive stress and parallel        to the maximum vertical compressive stress would enable at least        four elastic coefficients to be determined.

Referring to FIG. 100, under the two assumptions (TI axis parallel toborehole and TI axis perpendicular to borehole (FIGS. 9A, 10A and 9B,10B)), there are two values for PR and two values for YME, with PR andYME being elastic coefficients.

In practical application, the rock formation may not be actuallyhorizontal or vertical, but may be tilted. When drilling through titledmedia, particularly media with small deviated angles of less than 30degrees, given the expression of the trigonometric variables in relationto a rotation of the principle axis, the equations still produce usefulresults for PR and YME and may be used to inform the variation in themechanical rock properties and in particular inform the location offractures where the variation of the mechanical rock properties ispredicted to do so as is described below.

On the calculation of the elastic coefficients YME and PR through aMechanical Rock Property Analysis (MRPA) of the Method

FIG. 12 is a diagram illustrating linear stress strain relationshipswith curves fit to data pairs corresponding to different locations alonga bore hole. The slopes of the fit lines in these examples relate to YMEin different locations along a bore hole. To obtain data for populatingthe constitutive equation (or equations), various geophysical dataprocessing techniques are involved, where:

-   -   1. Sampling the MWD measurements at a sufficiently high        frequency to resolve a small degree of mechanical variability        (which may correspond to the nature and occurrence of a discrete        fracture a couple of mm wide), the measurements of the forces        acting on a rock formation in connection with a drilling bit,        (TOB and WOB) and fluid system pressure or annular pressure        (Ap), the angular speed of the bit expressed in revolutions per        unit of time (RPM) and the 3-components of motion that represent        the acceleration of the bit and where each of the measurements        taken in time corresponds to a discrete position of the bit        along the length wellbore (the MWD data)    -   2. Processing accelerations may be processed using geophysical        signal processing techniques to obtain (i) the average lateral        or rotary and vertical displacement of the bit that correspond        to a single revolution or turn of the bit and (ii) Root Mean        Squared (RMS) amplitude of the acceleration    -   3. Calculating the RMS averages of the TOB and WOB over time        windows that correspond to a single bit turn. Typical        penetration rates are usually 0.02 in/rev, while drilling at 240        rpm, would result in a single turn of the bit every 250 ms which        if sampled at 1 kHz would provide sufficient data to be able to        identify when the bit encounters a single-discrete fracture    -   4. Obtaining data pairs, which collectively define the linear        stress strain relationship, from MWD data using relationships        between the rms acceleration and displacement of the bit that        are appropriate for the loading conditions and motion of the bit        in relation to the axis of symmetry for the constitutive        equations used to describe the rock formation where it is        understood that the loading conditions are determined with        respect to the orientation of the drilling well.    -   5. Using curve fitting techniques to estimate the two parameters        needed to describe a line, the Slope and Intercept and        statistical descriptions of the variations of the two elastic        parameters YME and PR with respect to each of the linear        clusters as were identified with respect to the band-limited MWD        data that was used to generate the data pairs    -   6. Identifying where the distributions of the MWD data pairs        form spatially or temporally consistent clusters or otherwise a        locus of adjacent points that (i) can be described through the        application and use of curve fitting techniques in terms of        relationships that are linear and (ii) such that each of the        linear relationships can be used to determine the parameters of        line such as the slope and intercept in relation to the        mechanical rock properties YME and PR.

In one possible example, a set of data pairs are generated for theequation of FIG. 9A (10A) and/or the FIG. 9B (10B). The data pairs aregenerated along the length of a well bore. In one example, a data paircomprises (y, x) and the set of data pairs may be used to generate thelinear stress strain relationships illustrated in graphical form in FIG.12 for each of the cases shown in FIGS. 9A and 9B.

In one embodiment, the variables of the equations are populated using(i) the RMS acceleration data, which is used to describe the forcesacting on the formation in connection with the bit, and (ii) the ZFL ofthe displacement spectrum is used to describe the motion on the bitwhere the motion understood to be the strain experienced by the rockformation and where the orientations of strain are described by the bitdisplacements according to the cutting direction of the bit and theorientation of the borehole in relation to the orientation of the axisof material symmetry as is shown in Figure. Thus, the acousticalacceleration measurements may use to specify the forces acting on theformation. The approach essentially conforms with Newton's second lawand balance of forces.

Special Cases to Consider

As shown, the MWD parameters may be expressed as a function of thefrequency content. In particular, where the frequency content is limitedthrough the application and use of a band-pass filter to generateband-limited MWD data pairs to populate the terms and conditions of theconstitutive equations of linear elasticity, using band-limited MWD datato form frequency-dependent, data pairs corresponding to data thatgenerally describes the stresses acting on the material and thedeformation of the material to populate the terms and conditions of theconstitutive equations of linear elasticity. Using geophysical signalprocessing techniques to obtain zero-frequency levels (ZFL) of thedisplacement spectra where the ZFL corresponds of depth of cut perrevolution of the bit or the depth of cut per unit of time or the rateof penetration (ROP) in relation to the width of the bandpass filterused to window the MWD data (the band-limited MWD data). That is the ZFLis determined from a specified range of frequencies or is otherwisecalculated from bandlimited data.

Using the two parameters YME and PR and the statistical descriptions ofthe variations of the two parameters as can be obtained through thecurve fitting technique for each liner cluster to diagnose the drillingconditions for each of the depths as a function of the frequency used toform the data pairs.

Fracture Identification from the Elastic Coefficients

It is usually not known a priori what the appropriate angularrelationships between the axis of symmetry of the constitutive elasticequations used to describe the rock formation and the axis of symmetryof the wellbore are. In practice, most horizontal wells are drilledparallel to bedding and most vertical wells are drilled perpendicular tobedding. Further, in basins, most natural fractures are vertical, andthus most horizontal wells are drilled perpendicular to fractures andmost vertical wells drill parallel to fractures.

Referring again to FIGS. 9A and 9B, the MRPA technique is used topopulate the terms and conditions of the two assumptive cases where theaxis of symmetry of the constitutive media used to describe the rockformation are examined for the cases when (i) the axis of materialsymmetry is perpendicular to the axis of drilling and (FIG. 9A) (ii) theaxis of material symmetry is parallel to the axis of drilling (FIG. 9B).In an isotropic media, the determination of YME and PR from the MRPAanalysis for the two constitutive stress strain relationships willresult in equally or closely spaced values of YME and PR that tend totrack each other in space and time. In anisotropic media, the twocomputations of YME and PR as provided by the MRPA will deviate inpredictable ways that can be used to identify the nature and occurrencefractures in relation to (i) the differences in the elastic coefficientsas specified by the type of anisotropy, either HTI or TVI, that isencountered relative to the orientation of the drilling well and (ii)the reduction in strength of the rock as is provided by variations inHTI YME and the TVI YME.

In the case of a horizontal well when the constitutive equationsdescribe a variation in the end members situations (i) the rock may beunderstood to be fractured when PR TVI is lower than PR HTI and themagnitude of the fracturing is related to the decrease in thecalculation of YME. Typically, in practice, fractured zones can bediscerned in horizontal wells when the media is TVI PR is lower than theHTI PR. The difference in the values of the elastic coefficients YME andPR between the HTI and VTI represent two cases of a cross-over curve oran anisotropic cross-over curve.

In the case of a vertical well, the rock is understood to be fracturedwhen HTI PR is less than TVI PR (e.g., FIG. 13, discussed in more detailbelow). An objective way to determine these cross-over relationships (PRor YME for the two assumptive cases) is by calculating a series in timeor a series in depth (the logs) of the YME or the PR for both the TVIand HTI solutions. Time or depth may be correlated to length along thewell bore from which the measurements were made.

Mechanical rock property logs (the “Logs”) as calculated using theequations set out by the method can be smoothed by averaging the PR andYME values by (i) using the statistics in relation to the goodness offit to the curve where one such statistic is known as the R statistic asprovided by standard least-squares linear curve fitters such as LINESTin Excel, to filter out data with poor statistical evidence for a linearrelationship between the data pairs or (ii) to weight the values of thecurve at a particular location. Using a curve smoothing technique thatwould average the data over a time window where the length of the timewindow corresponded to the variation of the data and then resampling thetime window in either time or depth according to its position in thesubsurface. Processing logs using smoothing techniques can improve theability to identify the relative variations in the elastic coefficients.

Further processing the logs by subtracting the mean value from the HTIand TVI YME Logs where the mean value is a running average of the dataalong the Log where the length of the running average is related to:

-   -   (1) the measure of the smallest length of mechanical anisotropy        variation of interest which in terms of the practice and        application of the method may involve spatial distances as small        as 1 inch for the sample rate and frequency content that is        afforded by the application of use of state-of-the-art MWD in        relation to the method        -   AND    -   (2) of sufficient resolution to document the nature and        occurrence of changes in rock properties needed to identify        fractures at the resolution need for the commercial exploitation        of commercial hydrocarbons from unconventional reservoirs.

The subtraction of the mean value of from the HTI and TVI PR Logsprovides a baseline from which to compare the variations between thecurves in ways that can be used to identify the locations of fractures.In practice the mean values of the Logs can be calculated as the averageof the data values in the logs over a certain time or distance specifiedby the spatial position in the subsurface along the length of thewellbore from which the measurements were made. When this mean value issubtracted from the Logs it provides the mean-subtracted Logs from whichto make it convenient to obtain relative comparisons (e.g., PR HTI to PRVTI and/or YME VTI to YME HTI).

Taking differences of the mean-subtracted YME HTI Log and themean-subtracted YME TVI Log and taking the differences of themean-subtracted PR HTI log and the mean-subtracted PR TVI log providesan identification of the type of rock anisotropy based on theorientation of the well in relation to the axis of material symmetry.The differences in these Logs as evidenced by the behavior of theelastic coefficients in relation to the orientation of media symmetrywith respect to the orientation of the wellbore can be can be understoodin predictable ways to describe the location of a zone of weakness inrelation to the method understood to be a fracture. For the casespresented here, these relationships provide a predictable way toidentify fractured rock formations, among other advantages

Specific Case 1: Vertical Well, Horizontal TI Media

In typical TVI media, the vertical PR (FIG. 9A—TI axis parallel toborehole) typically has lower values than horizontal PR (FIG. 9B TI axisperpendicular to borehole). That is, the material is more compliant to aload that is applied perpendicular to the axis of material symmetry thanto a load applied parallel to the axis of material symmetry. Loading therock formation in the same direction as the axis of material symmetrywill result in less horizontal deformation, decreased horizontalcompliance and/or higher ratios of horizontal to vertical stiffness.Conversely, loading the rock formation perpendicular to the axis ofmaterial symmetry, when the material symmetry is governed by fractures,will have higher compliances, higher PR and lower YME.

Stated differently, in a vertical well when the media is layercake (theaxis of media symmetry is parallel the borehole), the weight on bit andaxial displacements are typically parallel to the axis of materialsymmetry. In this case the VTI PR is typically less than HTI PR.Therefore, detection of zones where PR VTI is greater than PR HTIimplies that the material behavior under the loading conditions of thebit is stiffer or less compliant in the horizontal direction as opposedto the vertical direction. Referring to FIG. 13, when a vertical wellencounters HTI media (in a formation expected to be VTI), themean-subtracted PR HTI log becomes less than the mean-subtracted PR TVIlog, and a Fracture ID flag may be generated. This flag means the ratioof the displacement parallel to the axis of material symmetry increasesrelative to the displacement perpendicular to the axis of materialsymmetry. This is evidenced through the application and use of MWD datato describe the constitutive behavior of a rock formation anddemonstrate an increase in PR TVI relative to PR HTI. This increase inPR evidences the presence of vertical fractures or a verticallyfractured rock formation when drilling a vertical well. This logic isgenerally true is most circumstances, because most vertical wells aredrilled perpendicular to the bedding planes of the rock formation wherethe well bore axis is parallel to the axis of material symmetry and sothe presence of high VTI PR values is probably not related to verticalbedding planes. This behavior of a vertically drilling well encounteringa set of vertical fractures can be further corroborated by a similarexamination of the differences between the VTI YME and the HTI YME.Because the torque on bit is acting as a body force parallel to the axisof material symmetry as would be expected in the case of a verticallydrilling well encountering a set of vertical fractures where thefractures control the axis of material symmetry, the VTI YME decreasesand the HTI YME increases.

When the YME TVI log crosses or decreases relative the YME/HTI log for ahorizontal well then it is likely a zone of weakness that is morecompliant in the direction of loading that is parallel to the WOB orperpendicular to the axis of material symmetry has been detected whichwould again be consistent with a zone of vertical fractures. So, in someinstances, PR crossover would generate a flag, YME crossover wouldgenerate a flag, and in some instances the presence of both flagsindicate a fracture. Moreover, in practice, a threshold may be appliedthat would need to be met before generating a flag. In one example, forPR data, a distribution curve may be generated for all positivecrossovers, and only crossovers exceeding 68th or 90th or 95thpercentile may be flagged. Other thresholds or data technique may alsobe used to eliminate data points that may be attributable to noise.

When the relationship between the HTI and TVI elastic constants returnsto a pre-crossover relationship for a vertical well in an HTVI media,the crossover Fracture ID flag is set back to zero. The flag may remainset, however, for as long as the log indicates. So, as shown in theexample of FIG. 13, there are six sections of the 100 foot illustratedborehole where fractures are identified over several feet for eachsection.

Specific Case 2: Horizontal Well, Horizontal TI Media

FIG. 14A illustrates the two YME computations over about 100 feet ofhorizontal well. FIG. 14B illustrates PR ratio computations over thesame 100 fee of horizontal well. The YME and PR computations are basedon data obtained while drilling. In this case, only the geophysicalprocessing of the acoustical data as obtained from the accelerometersare used to populate the equations, so not direct measurements of WOB orTOB are provide in this example, but they could be included in anotherapplication and use of the equations as provided by the method. In theexample illustrated in FIGS. 14A and 14B, a fracture ID may be generatedwhere shown. In this case, effectively the opposite behavior than wasdescribed for a vertical well drilling parallel to the axis of symmetryor HTVI or layercake, is expected because now, in the case of ahorizontal well in TI media, the axis of drilling is perpendicular tothe axis of media symmetry. In this case, the loading conditions (torqueon bit) are parallel to the axis of material symmetry conditions androtational displacements (revolutions of the bit) are also parallel tothe axis of material symmetry and HTI calculation will result in a lowerHTI PR.

In the case of a horizontal well encountering a vertical fracture, therelationship is expected to be similar to a vertical well drilling in aHTVI media—the PR HTI will decrease relative to the mean-subtracted PRVTI. More specifically, in the event with the mean-subtracted VTI PR fora horizontal well drilling in a layercake media is found to be less thanthe mean-subtracted HTI PR, then it is likely that the axis of materialsymmetry relative to the orientation of the drilling well and loadingconditions of the bit has been rotated by 90 degrees. This can occurwhen the axis material symmetry is defined by a set of verticalfractures being intersected by a horizontal well. Thus, as shown, forexample in FIGS. 14A and 14B, a fracture flag may be identified in theidentified areas, as well as possibly other areas.

Likewise, the mean subtracted TVI YME will increase relative to themean-subtracted HTI YME when the material symmetry is as defined by aset of vertical fractures because torque on bit will be loadingperpendicular to the axis of material symmetry. In this example thecross-overs are not always synonymous as can be expected for real rockwhere a continuum of mechanical rock properties will occur based on thenatural heterogeneity of a complex, natural system where it isunderstood that the detection of fractures from among various therelationships between these cross-overs is just one implementation amongothers.

Among the advantages of the method is to use the values of thedifferences in the elastic coefficients between the various types ofcross-over than can be expected from the method where in simple casesfracture flags identified by the simultaneous cross of both YME and PRcurves as is afforded by the method, to other cases of mechanical rockheterogeneity that is caused by the occurrence of only one curvecrossing over the other or vice versa. Here this may lead to additionalfracture classification schemes that would involve the cross-over of oneset of coefficients relative to the other.

Elastic coefficients and the variation in the elastic coefficients wherethe variation of the elastic coefficients is determined in oneparticular application though the differences as obtained by asubtraction and of the elastic coefficients for the HTI and VTIinstances as is specified by the orientation of the well in relation tothe axis of material symmetry. When the variations determined from thestress-strain relationships based on the geographical orientation of thewell relative to the bedding planes provide an indication of fracturesin accordance with the manner described here would provide usefulinformation for the design the emplacement of hydraulic fracturestreatments and the selection of hydraulic fracture initiation points.

Other Improvements as are Envisioned by the Method

In one embodiment, the wellbore is drilled laterally through anunconventional shale reservoir. The natural variations in the strengthof an unconventional shale reservoir could be viewed by plotting aplethora of stress-strain relationships that could be derived inassociation with every single turn of the bit (FIG. 8). This wouldcreate a scatter plot that can be analyzed by using statistical methodsto find significant relationships in the data that are used to classifythe nature of the rock deformation based on the groupings of themechanical rock property data on the stress-strain diagram.

These classified rock strength measurements may be indexed according totheir spatial location along the well bore trajectory. Classification ofthe type of rock deformation such as strong or weak, brittle or ductilebased on the mechanical rock properties and in particular the elasticcoefficients in the stress-strain diagrams when logged using the MWDsystem could be used to identify and select zones along the wellboreselection of hydraulic fracture initiation points for the emplacement ofhydraulic fractures.

Other stress-strain relationships can be developed in the mannerdescribed here where the forces acting the formation in connection withthe bit are the effective stress obtained by differencing the force andthe fluid pressures in the drilling system or the Torque acting on thebit. In another embodiment the orientation and geometry of the cuttersin relation to the WOB and Torque can be used to describe tractions thatare normal and tangential to the cutting face and be used to specifyadditional coefficients of elasticity as provided by the method. Thenormal traction can be modified by the drilling fluid pressures tocalculate effective normal stress acting on the cutting face of the rockformation. When the effective normal stress and shear stress can beprojected onto a fault place where the fault plane undergoesreactivation as evidenced by the methods here, when used in conjunctionwith a failure criterion, can provide critical information on thestate-of-stress in the reservoir.

Inclination Measurements as Provided by the ZFL Measurements

Referring again to FIG. 6, it is understood that changes in theinclination of the bit, detectible with MWD data in combinations of thevarious forms discussed herein, may also identify discontinuities alonga bore hole. More specifically, as illustrated, if the drill bitencounters a mechanical discontinuity or geological boundary, thecutting face of the drill bit may change its orientation is response tothe orientation and stresses acting on the heterogeneity. Instantaneouschanges in inclination of the drill bit relative to a long-term averageinclination can be obtained using a sensor or an array of sensors (e.g.axial and lateral or rotary accelerometers) configured to record andextract signals in relation to the three, independent spatial axes ofthe drilling vibrations. The magnitude of the deflection of theinclination of the drill bit relative to a long-term trend in thedirection of the drill bit depends on the orientation of the mechanicaldiscontinuity or geological boundary with respect to the cutting face ofthe drill bit and therefore the deflection is understood to indicate achange in the mechanical rock properties. If there is an indication of afracture based on the RMS levels of the other measurements, but nocorresponding deflection in the bit, then it is understood that theorientation of the mechanical discontinuity is perpendicular to theorientation of the well bore trajectory. The magnitude of the deflectionof the drill bit can be determined through a principle componentanalysis of the signals extracted from the drilling vibrations where thetime window used to obtain principle components of drilling motion ofthe signals may be normalized by the bit speed and where the principlecomponents can be expressed as changes in the rate of penetration (ROP)or the acceleration of the drill bit.

Discrete Microearthquake Detection Method to Identify a Fault

Referring now to FIG. 15 if the forces acting on the formation inconnection with the drill bit and drilling fluid system when conductingdrilling operations are sufficient to overcome the failure criteria of apre-existing fault, then the fault will slip or fail. Reactivation of afault or pre-existing fracture can be evidenced by extracting a signalfrom the drilling vibrations that is related to a microseismic eventwith attendant primary, compressional (P) and secondary, or shear (S)arrivals. In the case where the fault is perpendicular to the trajectoryof the wellbore the P-wave arrival is related the particle motionparallel to the axis of the drill string and the S- or transverse waveis the particle motion parallel to the lateral and torsion motion of thedrill string. Deviations of the particle motion of the P- and S-wavescan be used to determine the orientation of the fault relative to thetrajectory of the wellbore.

Reactivation of a fault is expected to create much larger signals aretypically expected from the acoustic emissions generated by thefracturing of a rock formation in relation to the cutting action of thebit. By looking at the instantaneous amplitude levels relative to along-term trend, where the temporal windows used to select theinstantaneous amplitudes are related to the bit speed and the long-termwindow is related to the spatial distribution of the faults in the rockformation, it is possible to identify the location where the bitencountered and crossed a fault.

In the special case of a fault reactivation while conducting drillingoperations, the sensors deployed on the bottomhole assembly act like anearthquake seismometer where the geophysical signal processingtechniques identify the discrete arrivals of P-waves and S-waves withorthogonal particle motions to detect the presence and reactivation of apre-existing fault. If the orientation of the fault with respect to theorientation and magnitude of the forces acting on the formation inconnection with the bit and drilling fluid system can be determined thenthis would enable a method to specify in 3-dimensions a failurecriterion of the fault.

Geosteering or Real-Time Applications of the Method

Stick-slip drilling behavior causes the bit speed, typically expressedin revolutions, per minute to increase or decrease according to thedistribution of forces acting on the formation in connection with thebit. The variations of the bit speed with respect to the forces used tofracture a rock formation to achieve optimum rates of penetration aretypically used to describe the efficiency of the drilling operation,where there is an optimum efficiency that maximizes the rate ofpenetration with respect to the forces acting on the bit

Techniques that can determine the deformation of a rock formation bydescribing formation fracturing while conducting drilling provide thatwhen these techniques are enabled in real-time implementations using MWDsystems and apparatus, they also can be used to “geosteer” thehorizontal well at the bit by maintaining the trajectory of the wellborein the hydrocarbon bearing zone that experiences fracturing generated bythe cutting action of the bit in relation to the forces acting on theformation in connection with the bit and drilling system that hasbearing on the mechanical rock properties that will enhance theeffectiveness of hydraulic fracture emplacements.

Example embodiments described herein regarding the various controlmethods may be implemented at least in part in electronic circuitry; incomputer hardware executing firmware and/or software instructions;and/or in combinations thereof. Example embodiments also may beimplemented using a computer program product (e.g., a computer programtangibly or non-transitorily embodied in a machine-readable medium andincluding instructions for execution by, or to control the operation of,a data processing apparatus, such as, for example, one or moreprogrammable processors or computers). A computer program may be writtenin any form of programming language, including compiled or interpretedlanguages, and may be deployed in any form, including as a stand-aloneprogram or as a subroutine or other unit suitable for use in a computingenvironment.

FIG. 16 is a flowchart illustrating one method conforming with aspectsof the present disclosure. It should be recognized that the elaboratedetail and various alternatives and embodiments set out herein mayconstitute other methods, alone or in combination with that set forth inFIG. 16. Moreover, various steps of the method of FIG. 16, as well asother methods, may be performed within a computer system such as set outin FIG. 17 or may be performed, in whole or in part, in a bottom holeassembly associated with or proximate a bit, such as shown in FIG. 2,may form or be used in steering and may therefore be deployed in thegeosteering system illustrated in FIG. 2 or may be deployed in variousmechanisms associated with completions. Referring to FIG. 16, the methodinvolves receiving acoustical signals obtained from one or more sensorspositioned on a component of a bottom-hole assembly (operation 1610).The sensors (e.g., accelerometers) are in operable communication with atleast one data memory to store the acoustical signals (e.g., vibrationdata) where the acoustical signals are generated from a drill bitinteracting with a rock formation while drilling a wellbore. The methodfurther involves processing the acoustical signals to obtain at leastone set of data values representative of a mechanical rock property ofthe rock formation along the wellbore created by the drill bitinteracting with the rock formation for a period of time (operation1620). After which, the method may involve identifying a change in theat least one set of data values, where the change is representative ofthe drill bit crossing a mechanical rock property discontinuity whiledrilling the wellbore (operation 1630). In some instances, the methodmay further involve using the data to complete the well (operation1640).

As will be appreciated from the devices, systems and methods providedand disclosed herein, aspects of the present disclosure may also involvethe determination of absolute values of mechanical rock properties. Thesystems and techniques involve the application and use of (i) the forcesor accelerations of the drill bit and (ii) the displacements or motionsof the drill bit. Such force or acceleration and displacement data maybe obtained from recording the near-bit mechanical drilling vibrationsin relation to the drill bit interacting with a material having knownproperties and with a rock formation. In one specific example, drillingvibrations experienced by the drill bit from its breaking of rock whiledrilling, propagate as acoustical signals that are translated into databy accelerometers or other sensors positioned proximate the drill bit.The acoustical signals are translated into mechanical rock propertiesaccording to the techniques discussed herein. Further, by first drillingthrough some known media with known mechanical rock properties, such asa cement 38 (see FIG. 2B) in the well, the system may capture thevibration data and generate scalars. In turn, the scalars can be used totransform derived mechanical rock properties, from data captured in thesame manner but for an unknown media, such as a formation being drilledthrough, into mechanical rock properties for that formation. In oneexample, the mechanical rock properties for the formation are consideredabsolute values in that the mechanical rock properties have beennormalized by first obtaining scalars for the mechanical rock propertycomputation, which may be based on using the same drill bit and relatedcomponents to drill through a media with known properties.

Stated differently, the present disclosure outlines an innovativetechnique, and associated systems and apparatus, to obtain at least oneset of calibration values, which in one specific case are scalars, inrelation to drilling a rock formation or material with known mechanicalproperties. Scalars obtained in the manner presented here provide for away to transform MWD data to stress and strain experienced by a rockformation or material when interacting with a drill bit.

An MWD device, as disclosed, measures and obtains data pertaining todrilling forces, such as the weight acting on the bit or the torqueacting on the bit (i.e., WOB and TOB) and/or accelerations that describethe angular and linear motions of the bit. Mechanical rock propertiesmay be described using stiffness coefficients, colloquially known as theCij's, that are expressed in terms of stress represented generally asforce per area and more specifically as pounds per square inch.

Stiffness coefficients of rock formations are expressed in terms offorce per unit area and are on the order 1e10 Pascal's or several Mpsi.Downhole measurements of forces such as weight on bit are on the orderof 10's of kilopounds and torque on bit on the order of a few kilopoundsper foot. Typical values of near-bit accelerations used to represent theforces acting on the formation can be on the order of several g's. And,typical values of the displacement of the bit as determined fromprocessing the near-bit accelerations are on the order of severalmicrometers.

Because the MWD data represent forces or accelerations, anddisplacements as opposed to stress (σ) and strain (ε), respectively,corrections that take into account the physics of the elastic radiationof the vibrations generated by the drill bit, the transmission ofvibrations as acoustical waves to the MWD data recorder 36, and thelengths and areas over which the radiation and transmission occur needto be obtained to transform the MWD data to units that can be used toobtain the stiffness coefficients or the material or rock formationbeing drilled. Likewise, the forces obtained from MWD data, such asweight on bit and torque on bit can be transformed to stresses with theunderstanding that a geometric correction factor in relation to theeffective contact area is also involved. One example of such a contactarea is the area of the bit.

In practical terms, it may be difficult or impossible to know, in-situ,the actual and precise contact area for any given turn of the bit. Thearea of the bit in contact with the rock formation depends on thevarious configurations of the cutters on the bit with respect to theweight on the bit and the wear of the cutters on the bit. More weight onthe bit presses the cutters further into the formation and results in anincrease in contact area as a function of the weight on bit and in acomplex fashion. Also, the loss and wear of cutters during normaldrilling operations will also have unpredictable and hard to determinechanges in the contact area of the bit.

The motion of the bit also needs to be transformed or otherwise scaledto a strain in order to process the data using a stress-strainrelationship and to obtain absolute values for the mechanical rockproperties. The displacement of the bit that occurs relative to a givensegment or reference length of the rock formation may be used to obtainthe strain experienced by the rock formation when interacting with adrill bit. One such example of the reference length of the rock could bethe circumference of the borehole. Using the circumference of the bitused to obtain the reference length would have to account for changeswith respect to the radius of the bit or possibly in an arbitrary mannerdepending on the geometry and configurations of the cutters on the bit.

The use of geometrical constructs and analytical expressions of wavegeneration and propagation to transform the MWD data to obtain stressand strain of a rock formation interacting with a drill bit isunsatisfactory because, (i) the reference lengths of the rock withrespect to the displacement of the bit is generally unknown, (ii) thearea of the bit is rugose and variable, and (iii) the radiation andscattering of energy from the bit rock interface and the transmission ofthe energy to the MWD recorder is difficult if not impossible toaccurately predict in-situ and under changing drilling conditions.

Referring to FIG. 17, here we disclose an innovative method, whichoperations are performed by computing system, including a processor, toobtain a set of scalars from drilling a rock formation or material withknown mechanical properties that transforms the accelerations or forces,and displacements obtained from MWD data to corresponding stresses andstrains. The method involves (i) drilling through a material with knownmechanical properties and obtaining signals, such as acoustical signalsfrom the drill bit interacting with the material (operation 1710), (ii)processing the MWD data to obtain the forces on the bit and the motionsof the bit interacting with the material (operation 1720), (iii)populating a general stress-strain relationship using (a) the forces onthe bit, (b) the motions of the bit, (c) the values of the knownmechanical rock properties and (d) the scalars (unknown) that are to bedetermined according to each force and displacement (operation 1730),and (iv) obtaining values of the scalars that satisfy conditions of thestress-strain relationship in relation to the knowledge of themechanical rock properties (operation 1740). The acoustical signals maybe stored in memory of the bit sub. These scalars are understood toaccount for geometrical considerations such as the effective areas andreference lengths in relation to the bit design and take intoconsiderations other complex, hard-to-predict factors, such as theradiation and transmission of the drilling induced vibrations from thecutters to the MWD sensors.

The techniques discussed herein involve processing a material havingknown mechanical rock properties. Knowledge of the mechanical rockproperties, which may be considered absolute values, useful to processdrilling vibrations in relation to the method described can be obtainedindependently from common well known methods such as from (i) sonic oracoustical measurements of rock velocities that can be systematicallyrelated to mechanical rock properties or (ii) the application and use oftechniques that systematically measure the deformation of a rock sampleor a core from a rock sample in response to a given load or stress toobtain mechanical rock properties. Other knowledge about absolute valuesof mechanical rock properties that can be used to scale drillingvibrations may also be obtained from computer generated models of rockmineral stiffness coefficients. In some instances, the absolute valuesof mechanical rock needed to scale the drilling vibrations can bearrived at through deductive or experiential means or what is commonlyunderstood as a best guess. Cement is one example of a material withknown mechanical properties that is commonly encountered in a wellboreduring the drilling process.

The process of drilling a horizontal well may be completed in a seriesof stages. The first stage may involve the drilling of a vertical wellto a target formation 30, also referred to as a hydrocarbon bearingformation, which is part of a larger reservoir. After the first stagehas been drilled, the wellbore is typically cased with steel tubes,which are held or suspended in place by cement between the outside ofthe steel and the wellbore. After setting the casing, there is typicallycement 38, referred to as a cement plug, left in the bottom of the well.The second stage would then involve sidetracking to a lateral ordrilling through the cement plug in order to begin drillingdirectionally forming a horizontal well 38 towards the reservoir inorder to intersect and target hydrocarbon formations of the reservoir.

In the instance where the rock formation or material being drilledincludes a cement plug with known mechanical rock properties per orgiven the constituents of the cement or other techniques that can beused to investigate the mechanical properties of a cement as describedabove, then the processing of the acoustical signals generated bydrilling vibrations involves an innovative way to obtain stress andstrain from corresponding forces or accelerations of the bit, and themotions or displacements of the bit in relation a set of scalars (A, B,C, D, E and F) with respect to or representative of the known mechanicalrock properties.

The mechanical properties of the cement can be expressed as an isotropicmedia in terms of two elastic stiffness coefficients C₁₁ and C₁₂. Theisotropic stiffness coefficients and the forces or accelerations anddisplacements of the bit in relation to the scalars (A, B, C, D, E, F)that are required to transform the MWD data to stress and strain can bearranged through a system of linear equations as follows:

${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$

This stress-strain relationship represents a homogeneous system oflinear equations that can be solved for the scalars (A, B, C, D, E andF) from measurements while drilling a cement or other isotropic materialwhere the mechanical properties are known. Here the displacements or “d”terms and the accelerations or “a”, terms are specified respectively interms of (i) the axial, angular, centripetal and/or lateral displacementof the drill bit and (ii) the axial, angular, centripetal and/or lateralaccelerations of the bit when interacting with the cement. Values forthe stiffness coefficients, C₁₁ and C₁₂, are known for the sample media.In various possible examples, the “d” terms may be any combination ofdisplacements (e.g., d₁ is axial, d₂ is angular, and d₃ is centripetal,or d₁ is angular, d₂ is centripetal and d₃ is axial, or d₁ and d₂ areaxial and d₃ is angular, etc.), and the acceleration or the “a” termsmay similarly be any combination of accelerations.

It would also be possible to represent the “a” terms using dynamic data,such as from strain gauges associated with the MWD, of the WOB and theTOB where the solution to the scalars would represent the effectivecontact areas of the bit with respect to the orientation of the forcesbeing applied to the bit.

Stated independently, the effective contract areas of the bit acted onby the torque is not expected to be the same as the effective contactarea acted on by the weight on bit. The differences in these contactareas depend on the geometry and configuration or arrangement of thecutters and the exposure of the cutters. In the case of scaling theweight and torque forces to stress, the scalars describe how the cutterexposure with respect to a particular bit design transform the forces tothe stresses normal and perpendicular to the motion of the bit.

The method can also be extended to include the cases when the forces onthe bit are obtained from surface drilling dynamics measurements, suchas WOB and TOB, where the torque is obtained from surface measurementssuch as the rate of fluid flow through the motor.

The values of the scalars can also be obtained, in relation to othermore exotic material symmetries as well such as transverse isotropy,orthotropy, triclinicity, etc.

Let it also be understood that any assumptions placed on the scalarssuch as B=C or A=1 or B=0 represent degenerative cases of the statedsolution.

This system of equations can be modified to obtain scalars using otherisotropic representations of the elastic coefficients, where thestress-strain relationship may be expressed using elastic coefficientssuch as Young's modulus of elasticity and Poisson's ratio and thereforethe formulation as shown above in terms of the stiffness coefficientsC₁₁ and C₁₂ should not be considered limiting.

Because the cement plugs can be of hundreds of feet in length, drillingthis cement casing plug with respect to time may take several minutes oreven hours to drill. Therefore, MWD data obtained from of a drill bitinteracting with cement may result in multiple measurements of theforces or accelerations, and displacements allowing the population ofthese equations under different drilling conditions and therebydetermining a single set of scalars that satisfy a solution to thehomogeneous stress-strain equations.

The application and use of a single set of scalars to transform MWD datato stress and strain can be thought of like this: if the cementproperties along the length of the lateral are constant and isotropic,and the drilling conditions vary, which they will, then the variationsin the stress and strain experienced by the cement that are caused bythe variations in the applied drilling behavior, are understood to fallalong a locus of points that defines or otherwise describes the elasticportion of the stress-strain relationship as is given the isotropicstiffness coefficients of the cement or other isotropic material withknown mechanical properties.

The homogeneous system of equations can be solved using algebraictechniques to construct the inverse of a matrix, e.g., by using asingular value decomposition (SVD). The SVD describes a family ofsolutions to the given system of equations in terms of linearcombinations of the eigenvectors that correspond to the zero valuedeigenvalues of the SVD decomposition. The set of scalars obtained from asolution to the homogeneous equations is given by a linear combinationof the eigenvectors, (U, V and W) associated with the zero-valued eigenvalues as follows:

${{\rho_{1}\begin{bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4} \\u_{5} \\u_{6}\end{bmatrix}} + {\rho_{2}\begin{bmatrix}v_{1} \\{\; v_{2}} \\v_{3} \\v_{4} \\v_{5} \\v_{6}\end{bmatrix}} + {\rho_{3}\begin{bmatrix}w_{1} \\w_{2} \\w_{3} \\w_{4} \\w_{5} \\w_{6}\end{bmatrix}}} = \begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}$

A solution for the scalars is obtained by ρ₁=1 and ρ₂=ρ₃=0.

While cement can be represented as an isotropic material, othermeasurements, like acoustical measurements or sonic measurements of rockvelocity can be used to specify, a priori, the absolute values of themechanical rock properties in terms of stiffness coefficients or otherelastic coefficients for a transverse isotropic media or other moreexotic media symmetries. In the case of a transverse isotropic media,the stress-strain equations in relation to the scalars are:

${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$

This stress-strain relationship represents a homogeneous system oflinear equations that can be solved for the scalars (A, B, C, D, E andF) from measurements while drilling cement. Here the displacements (“d”terms) and the accelerations (“a” terms) are specified respectively interms of (i) the axial, angular, centripetal and/or lateral displacementof the drill bit and (ii) the axial, angular, centripetal and/or lateralaccelerations of the bit when interacting with the cement. Unlike theisotropic case, for transverse isotropic media, it should be appreciatedthat the terms for the accelerations and displacements are arrangedaccording to the orientation of the drilling well with respect to theaxis of material symmetry. For example, laterally drilling a verticallytransverse isotropic (VTI) media would place the axial acceleration andaxial displacement with respect to the a₁ and d₁ terms.

It would also be possible to represent the “a” terms using MWD of theweight on bit and the torque on bit where the solution to the scalarswould represent the effective contact area of the bit. Again, forlaterally drilling a VTI media the weight on bit measurement would beused to inform the a₁ term and the scalar A, would be understood toreference the effective area of the cutter exposure acting in thedirection of drilling. The scalars obtained in this manner describe howthe geometry and configuration of the cutters convert the drillingforces to stresses.

The solutions to the scalars in this example would follow as for theisotropic example where a linear combination of the three eigenvectorsassociated with the zero eigenvalues obtained from a SVD of thecoefficient matrix satisfies the homogenous equations and can be used toobtain the scalar values.

We have disclosed a method to obtain a set of scalars that when appliedto the motions and accelerations or forces of a drill bit interactingwith a rock formation or other material with known values of mechanicalproperties, and, in particular, a cement, can be used to transform MWDdata to obtain stress and strain experienced by the material or rockformation.

In another rendition of the methods discussed herein, a computer modelmay be used to describe the stresses and strains of a material withknown mechanical properties interacting with a drill bit. These stressesand strains can be obtained using a computer aided drafting of thearrangement and geometries of the cutters in contact with a computerdescription of an elastic material undergoing elastic deformation whensubjected to computer generated forces and motions of the bit. Theforces and motions of the bit can be converted to stress and strain withrespect to the arrangement and geometry of the cutters on the bit.Accordingly, when actually drilling a material with unknown propertiesbut with the same forces and displacements or stresses and strains asinformed by the computer model, those stresses and strains can be usedto scale the MWD data and in particular the accelerations anddisplacements directly to obtain the scalars according to:

${A = \frac{\sigma}{a_{1}}},{B = \frac{\tau}{a_{2}}},{C = \frac{\tau}{a_{3}}},{D = \frac{ɛ_{axial}}{d_{1}}},{E = \frac{ɛ_{radial}}{d_{2}}},{F = \frac{ɛ_{angular}}{d_{3}}}$

where:

-   -   the σ and t terms represent one of a stress acting perpendicular        or parallel to the axis of drilling and the ε terms represent        the axial or one of the angular or linear strains perpendicular        to the axis of drilling, and    -   the “a” terms represent the acceleration of the bit with respect        to the axis of drilling and the “d” terms represent the        displacements of the bit with respect to the axis of drilling.        Note, the motions and accelerations of the bit are scaled with        respect to the corresponding stresses and strains obtained from        the computer model of the bit interacting with the rock        formation to obtain meaningful transformations of the MWD data        to stress and strain.

Next, we further elaborate on the method through the application and useof the scalars to (i) not only transform the MWD data to stresses andstrains but also, (ii) to obtain a general stress-strain relationshipwhen drilling a rock formation with unknown mechanical rock properties(operation 1750). Using the scalars to transform the MWD data to stressand strain, the isotropic stress-strain relationship is described asfollows:

${\begin{bmatrix}{Dd}_{1} & ( {{Ed}_{2} + {Fd}_{3}} ) \\{Ed}_{2} & ( {{Dd}_{1} + {Fd}_{3}} ) \\{Fd}_{3} & ( {{Dd}_{1} + {Ed}_{2}} )\end{bmatrix}\begin{bmatrix}C_{11} \\C_{12}\end{bmatrix}} = \begin{bmatrix}{Aa}_{1} \\{Ba}_{2} \\{Ca}_{3}\end{bmatrix}$

FIG. 18 is a diagram depicting the path of an accelerometer 1802 as adrill bit bores a hole along an axis of drilling 1804. The accelerometermay be positioned on or otherwise form a part of an MWD assembly. In anyevent, the accelerometer is positioned proximate a drill bit. Thediagram is an isometric view of a path of the accelerometer as the drillbit rotates clockwise 1808. In this representation, the axial (d₁),lateral (d₂) and centripetal (d₃) displacements when multiplied by therespective scalars, D, E, F are the axial, lateral and centripetalstrains, and likewise the axial (a₁), lateral (a₂) and centripetal (a₃)accelerations when multiplied by the respective scalars, A, B, C are theaxial, lateral and centripetal stresses.

This system of equations can be solved using well known algebraictechniques to obtain a solution for the isotropic stiffness coefficients(e.g., C₁₁ and C₁₂). Because, the MWD data have been transformed tostress and strain through the application and use of the scalars, theisotropic stiffness coefficients obtain in this manner representabsolute values of the mechanical rock properties.

Note, the coefficients can be applied to obtain the absolute values ofmechanical rock properties for other media symmetries such as transverseisotopic media with the media being horizontally transverse isotropic(HTI) or VTI as specified by the axis of material symmetry in relationto the orientation of the drilling well, such as follows:

${\begin{bmatrix}{Dd}_{1} & {Ed}_{2} & {Fd}_{3} & 0 \\{Ed}_{2} & {Dd}_{1} & {Fd}_{3} & 0 \\0 & 0 & ( {{Dd}_{1} + {Ed}_{2}} ) & {Fd}_{3}\end{bmatrix}\begin{bmatrix}C_{11} \\C_{12} \\C_{13} \\C_{33}\end{bmatrix}} = \begin{bmatrix}{Aa}_{1} \\{Ba}_{2} \\{Ca}_{3}\end{bmatrix}$

This method further advances the application and use of the absolutevalues of the elastic coefficients in predictable and systematic ways toinform the identification of fractures, bedding planes, materialboundaries or other discontinuities that act to separate or other offsetrock formations with different mechanical rock properties with respectto axis of media symmetry in relation to the orientation of the drillingwell as follows.

Lateral Heterogeneity Along a Wellbore

The application and use of absolute values of the stiffness coefficientscan be applied to in predictable ways to represent material or rockformation heterogeneity. In this instance, the presence of lateralheterogeneity encountered while drilling a well, and in particular wherethe lateral heterogeneity is generated through the nature and occurrenceof fractures or a dilute system of flat aligned microcracks, then theidentification of the lateral heterogeneity can be understood bycomparing the absolute values of the isotropic stiffness coefficients tothe stiffness coefficients that would understood to occur through theintroduction of a set of parallel slip interfaces that are understood tosimulate the presence of fractures or a fracture system as a set ofclosely spaced aligned joints. In the specific case, when the fracturesare flat and have rotational symmetry, then the effect of introducingthe fractures on an isotropic media may be specified through theintroduction of two additional elastic parameters that effectivelychange the representation of the media, under the aforementionedassumptions to become transversely isotropic.

Under the methods presented here, where the scalars can be used todetermine the stress-strain relationships and obtain absolute values ofthe stiffness coefficients for both an isotropic and a transverselyisotropic media, advances previous techniques to identify the presenceof fractures based on predictable differences of the stiffnesscoefficients between the isotropic and transversely isotropic media withrespect to the orientation of a drilling well.

This is an improvement upon previous techniques where the differencesbetween elastic coefficients obtained with respect to the orientation ofthe material symmetry in relation the axis of drilling were specifiedaccording to the relative changes of the elastic rock property curves.

Drill Bit Specification

The processing and analysis of drilling induced vibrations, under themethods disclosed here, can be used to describe how the arrangement andgeometry of the cutters on a drill bit transform the forces oraccelerations, and motions of the bit to the stresses and strainsexperienced by a rock formation or material. That is, the scalars can beused to specify the performance of a specific bit design in terms of thestresses and strains. The scalars therefore can be used to select orotherwise specify a bit design that is able to generate sufficientstresses necessary to cause the deformation and failure of a material orrock formation.

Because rock fragments according to the stresses experienced by a rockformation when interacting with a drill bit, using the scalars to obtainthe stresses from MWD data can be used to improve the design of the bitwith respect to the configuration and geometries of the cutters wheninteracting with a material or rock formation with known mechanicalproperties.

Under the scenario envisioned by the method, the processing and analysisof drilling induced vibrations to obtain stress and strain could be usedto (i) select the configuration and geometries of the cutters that wouldmaterially improve the cutting efficiency of the bit, and (ii) predictthe ability of the bit to drill through a material or rock formation ormore specifically the rate of penetration with respect to the mechanicalproperties that are likely to be encountered. Selecting a bit designthat maximized the rate of penetration with respect to the mechanicalrock properties to be encountered when drilling a well is an important.There is a need to improve techniques to select the optimal bit designto maximize the drilling rate of penetration. Therefore, the scalars canbe used with respect to the drilling forces to improve drillingperformance and pre-drill drilling performance by selecting a drill bitbased on the stress and strain that will be needed to drill a materialor rock formation to the character of what is being drilled. Forexample, when the drilling program is expected to drill through aparticularly hard or strong formation, then drilling performance may beenhanced by selecting a bit the generates more stress for a given rigcapability (available power and drill string specifications) oraugmentation through the introduction of a downhole motor in the BHA.This is useful for optimizing equipment selection, and in particular toselect a downhole motor develop a field given the mechanical rockproperties of the formations and reservoirs to be drilled.

FIG. 19 below is a block diagram of a machine in the example form of acomputer system 1900 within which instructions 1906 for causing themachine to perform any one or more of the methodologies, and variouscombinations of the same, discussed herein may be executed by one ormore hardware processors 1902. In various embodiments, the machineoperates as a standalone device or may be connected (e.g., networked) toother machines. In a networked deployment, the machine may operate inthe capacity of a server or a client machine in server-client networkenvironment, or as a peer machine in a peer-to-peer (or distributed)network environment. Further, while only a single machine isillustrated, the term “machine” shall also be taken to include anycollection of machines or controllers that individually or jointlyexecute a set (or multiple sets) of instructions to perform any one ormore of the methodologies discussed herein.

As depicted in FIG. 19, the example computing system 1900 may includeone or more hardware processors 1902, one or more data storage devices1904, one or more memory devices 1908, and/or one or more input/outputdevices 1910. Each of these components may include one or moreintegrated circuits (ICs) (including, but not limited to,field-programmable gate arrays (FPGAs), application-specific ICs(ASICs), and so on), as well as more discrete components, such astransistors, resistors, capacitors, inductors, transformers, and thelike. Various ones of these components may communicate with one anotherby way of one or more communication buses, point-to-point communicationpaths, or other communication means not explicitly depicted in FIG. 19.Additionally, other devices or components, such as, for example, variousperipheral controllers (e.g., an input/output controller, a memorycontroller, a data storage device controller, a graphics processing unit(GPU), and so on), a power supply, one or more ventilation fans, and anenclosure for encompassing the various components, may be included inthe example computing system 200, but are not explicitly depicted inFIG. 19 or discussed further herein.

The at least one hardware processor 1902 may include, for example, acentral processing unit (CPU), a microprocessor, a microcontroller,and/or a digital signal processor (DSP). Further, one or more hardwareprocessors 1902 may include one or more execution cores capable ofexecuting instructions and performing operations in parallel with eachother. In some instances, the hardware processor is within the bit sub,and others it is part of another separate processing system.

The one or more data storage devices 1904 may include any non-volatiledata storage device capable of storing the executable instructions 1906and/or other data generated or employed within the example computingsystem 1900. In some examples, the one or more data storage devices 1904may also include an operating system (OS) that manages the variouscomponents of the example computing system 1900 and through whichapplication programs or other software may be executed. Thus, in someembodiments, the executable instructions 1906 may include instructionsof both application programs and the operating system. Examples of thedata storage devices 1904 may include, but are not limited to, magneticdisk drives, optical disk drives, solid state drives (SSDs), flashdrives, and so on, and may include either or both removable data storagemedia (e.g., Compact Disc Read-Only Memory (CD-ROM), Digital VersatileDisc Read-Only Memory (DVD-ROM), magneto-optical disks, flash drives,and so on) and non-removable data storage media (e.g., internal magnetichard disks, SSDs, and so on).

The one or more memory devices 1908 may include, in some examples, bothvolatile memory (such as, for example, dynamic random access memory(DRAM), static random access memory (SRAM), and so on), and non-volatilememory (e.g., read-only memory (ROM), flash memory, and the like). Inone embodiment, a ROM may be utilized to store a basic input/outputsystem (BIOS) to facilitate communication between an operating systemand the various components of the example computing system 1900. In someexamples, DRAM and/or other rewritable memory devices may be employed tostore portions of the executable instructions 1906, as well as dataaccessed via the executable instructions 1906, at least on a temporarybasis. In some examples, one or more of the memory devices 1908 may belocated within the same integrated circuits as the one or more hardwareprocessors 1902 to facilitate more rapid access to the executableinstructions 206 and/or data stored therein.

The one or more data storage devices 1904 and/or the one or more memorydevices 1908 may be referred to as one or more machine-readable media,which may include a single medium or multiple media that store the oneor more executable instructions 1906 or data structures. The term“machine-readable medium” shall also be taken to include any tangiblemedium that is capable of storing, encoding, or carrying instructions1906 for execution by the machine and that cause the machine to performany one or more of the methodologies of the present invention, or thatis capable of storing, encoding, or carrying data structures utilized byor associated with such instructions 1906.

The input/output devices 1910 may include one or more communicationinterface devices 1912, human input devices 1914, human output devices1916, and environment transducer devices 1918. The one or morecommunication interface devices 1912 may be configured to transmitand/or receive information between the example computing system 1900 andother machines or devices by way of one or more wired or wirelesscommunication networks or connections. The information may include datathat is provided as input to, or generated as output from, the examplecomputing device 1900, and/or may include at least a portion of theexecutable instructions 1906. Examples of such networks or connectionsmay include, but are not limited to, Universal Serial Bus (USB),Ethernet, Wi-Fi®, Bluetooth®, Near Field Communication (NFC), and so on.One or more such communication interface devices 1910 may be utilized tocommunicate one or more other machines, either directly over apoint-to-point communication path or over another communication means.Further, one or more wireless communication interface devices 1912, aswell as one or more environment transducer devices 1918 described below,may employ an antenna for electromagnetic signal transmission and/orreception. In some examples, an antenna may be employed to receiveGlobal Positioning System (GPS) data to facilitate determination of alocation of the machine or another device.

In some embodiments, the one or more human input devices 1914 mayconvert a human-generated signal, such as, for example, human voice,physical movement, physical touch or pressure, and the like, intoelectrical signals as input data for the example computing system 1900.The human input devices 1914 may include, for example, a keyboard, amouse, a joystick, a camera, a microphone, a touch-sensitive displayscreen (“touchscreen”), a positional sensor, an orientation sensor, agravitational sensor, an inertial sensor, an accelerometer, and/or thelike.

The human output devices may convert electrical signals into signalsthat may be sensed as output by a human, such as sound, light, and/ortouch. The human output devices 1916 may include, for example, a displaymonitor or touchscreen, a speaker, a tactile and/or haptic outputdevice, and/or so on.

The one or more environment transducer devices 1918 may include a devicethat converts one form of energy or signal into another, such as from anelectrical signal generated within the example computing system 1900 toanother type of signal, and/or vice-versa. Further, the transducers 1918may be incorporated within the computing system 1900, as illustrated inFIG. 19, or may be coupled thereto in a wired or wireless manner. Insome embodiments, one or more environment transducer devices 1918 maysense characteristics or aspects of an environment local to or remotefrom the example computing device 1900, such as, for example, light,sound, temperature, pressure, magnetic field, electric field, chemicalproperties, physical movement, orientation, acceleration, gravity, andso on. Further, in some embodiments, one or more environment transducerdevices 1918 may generate signals to impose some effect on theenvironment either local to or remote from the example computing device1900, such as, for example, physical movement of some object (e.g., amechanical actuator), receiving or processing accelerometer data, straingauge data, and the like.

What is claimed:
 1. A method of characterizing rock propertiescomprising: receiving, at a processor, a first set of acoustical signalsobtained from one or more sensors, the acoustical signals generated froma drill bit interacting with a rock formation while drilling a wellbore;processing the first set of acoustical signals to obtain forces actingon the drill bit interacting with the rock formation while drilling thewellbore and to obtain displacements of the drill bit interacting withthe rock formation while drilling the wellbore; scaling the obtainedforces acting on the drill bit interacting with the rock formation whiledrilling the wellbore and the obtained displacements of the drill bitinteracting with the rock formation while drilling the wellbore toobtain information representative of stresses and strains of the rockformation; and processing the scaled forces and the scaled displacementsto obtain at least one set of data values representative of a mechanicalrock property of the rock formation along the wellbore created by thedrill bit interacting with the rock formation for a period of time. 2.The method of claim 1 wherein scaling further comprises: applyingscalars to the obtained forces acting on the drill bit interacting withthe rock formation while drilling the wellbore and the obtaineddisplacements of the drill bit interacting with the rock formation whiledrilling the wellbore, the scalars derived from acoustical signalsgenerated from the drill bit acting on a sample with known mechanicalrock properties.
 3. The method of claim 2 further comprising: obtainingthe scalars by: receiving a second set of acoustical signals obtainedfrom the one or more sensors wherein the one or more sensors arepositioned on a component of a bottom hole assembly, the second set ofacoustical signals generated from the drill bit interacting with thesample; processing the second set of acoustical signals to obtain forcesacting on the drill bit interacting with the sample and to obtaindisplacements of the drill bit interacting with the sample; processingthe obtained forces acting on the drill bit interacting with the sampleand the obtained displacements of the drill bit interacting with thesample to obtain the scalars that conform to a stress strainrelationship of the sample with known mechanical rock properties.
 4. Themethod of claim 3 wherein the stress strain relationship is:${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are forces or accelerations acting onthe drill bit interacting with the sample; the d values aredisplacements the drill bit interacting with the sample; where C₁₁ andC₁₂ are populated with known rock properties of the sample; and A-F arethe at least one set of scalars being obtained.
 5. The method of claim 4where the stress strain relationship is: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are forces or accelerations acting onthe drill bit interacting with the sample with respect to an axis ofmaterial symmetry of the sample; the d values are displacements thedrill bit interacting with the sample with respect to the axis ofmaterial symmetry of the sample; where the Cijs are populated with knownrock properties of the sample; and A-F are the at least one set ofscalars being obtained.
 6. The method of claim 4 wherein: the first setof acoustical signals are captured at the one or more sensors as axialaccelerations of the drill bit generated from the drill bit interactingwith the rock formation while drilling the wellbore and a lateral orrotary acceleration of the drill bit generated from the drill bitinteracting with the rock formation while drilling the wellbore; and thesecond set of acoustical signals are captured at the one or more sensorsas axial accelerations of the drill bit generated from the drill bitinteracting with the sample and a lateral or rotary acceleration of thedrill bit generated from the drill bit interacting with the sample. 7.The method of claim 6 wherein: processing the obtained forces acting onthe drill bit interacting with the sample and the obtained displacementsof the drill bit interacting with the sample to obtain the scalars thatconform to a stress strain relationship of the sample with knownmechanical rock properties wherein the stress strain relationship is afirst stress strain relationship of: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: d₁, d₂, and d₃ are any one of axial, centripetal,or lateral displacement of the drill bit during the period of time, a₁,a₂, a₃ may be any one of axial, centripetal or lateral acceleration ofthe drill bit during the period of time, C₁₁ and C₂₂ are known valuesfor the sample, and A-F are the obtained at least one set of scalarsfurther wherein: scaling the obtained forces acting on the drill bitinteracting with the rock formation while drilling the wellbore and theobtained displacements of the drill bit interacting with the rockformation while drilling the wellbore to obtain informationrepresentative of stresses and strains of the rock formation; andprocessing the scaled forces and the scaled displacements to obtain atleast one set of data values representative of a mechanical rockproperty of the rock formation along the wellbore created by the drillbit interacting with the rock formation for a period of time comprisesapplying a second stress strain relationship of: ${\begin{bmatrix}{Dd}_{1} & ( {{Ed}_{2} + {Fd}_{3}} ) \\{Ed}_{2} & ( {{Dd}_{1} + {Fd}_{3}} ) \\{Fd}_{3} & ( {{Dd}_{1} + {Ed}_{2}} )\end{bmatrix}\begin{bmatrix}C_{11} \\C_{12}\end{bmatrix}} = \begin{bmatrix}{Aa}_{1} \\{Ba}_{2} \\{Ca}_{3}\end{bmatrix}$ where: A-F are the obtained scalars of the first stressstrain relationship, d₁, d₂, and d₃ are any one of axial, centripetal,or lateral displacement of the drill bit as applied in the first stressstrain relationship to the rock formation by the drill bit during theperiod of time, a₁, a₂, a₃ are any one of axial, centripetal or lateralacceleration of the drill bit as applied in the first stress strainrelationship to the rock formation by the drill bit during the period oftime, and C₁₁ and C₁₂ are the obtained at least one set of data valuesrepresentative of the mechanical rock property of the rock formationalong the wellbore.
 8. The method of claim 4 wherein: processing thefirst set of acoustical signals comprises obtaining a root mean squareof the axial acceleration of the drill bit generated from the drill bitinteracting with the rock formation while drilling the wellbore and aroot mean square of at least one of the lateral and rotary accelerationof the drill bit generated from the drill bit interacting with the rockformation while drilling the wellbore, and obtaining an axialdisplacement of the drill bit generated from the drill bit interactingwith the rock formation while drilling the wellbore and at least one ofa lateral and rotary displacement of the drill bit generated from thedrill bit interacting with the rock formation while drilling thewellbore; and processing the second set of acoustical signals comprisesobtaining a root mean square of the axial acceleration of the drill bitgenerated from the drill bit interacting with the sample and a root meansquare of at least one of a lateral and rotary acceleration of the drillbit generated from the drill bit interacting with the sample, andobtaining an axial displacement of the drill bit and at least one of alateral and rotary displacement of the drill bit generated from thedrill bit interacting with the sample.
 9. The method of claim 2 whereinthe sample is cement used to set a casing of the wellbore.
 10. Themethod of claim 2 wherein the sample is a rock formation with knownmechanical rock properties.
 11. The method of claim 2 wherein the knownmechanical rock properties of the sample are obtained from at least oneof sonic measurements, core measurements, cutting measurements, seismicmeasurements, wireline log measurements, and a mineralogical model. 12.The method of claim 1 wherein the set of data values include at leastone absolute Cij value.
 13. A method of calibrating mechanical rockproperty derivations from a drilling tool comprising: obtaining, at aprocessor, a force data and a displacement data from signals obtainedfrom one or more sensors positioned on a component of a drilling toolproximate a drill bit, the force data and the displacement data being ofa drill bit interacting with a material with known mechanical rockproperties; processing the force data and the displacement data toobtain a set of scalars that conform to a stress strain relationship ofthe material with a known mechanical rock property.
 14. The method ofclaim 13 wherein the stress strain relationship is: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are the force data with respect to anaxis of material symmetry of the material; the d values are thedisplacement data with respect to the axis of material symmetry of thematerial; where C₁₁ and C₁₂ are populated with the known rock propertyof the material; and A-F are the set of scalars.
 15. The method of claim13 wherein the stress strain relationship is: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are the force data with respect to anaxis of material symmetry of the material; the d values are thedisplacement data with respect to the axis of material symmetry of thematerial; where the Cijs are populated with the known mechanical rockproperty of the material; and A-F are the set of scalars.
 16. The methodof claim 13 wherein the known mechanical rock property is obtained fromat least one of sonic measurements, core measurements, cuttingmeasurements, seismic measurements, wireline log measurements, and amineralogical model.
 17. The method of claim 13 wherein the material iscement.
 18. The method of claim 13 wherein the signals are acousticalsignals, the one or more sensors are at least one of an accelerometersand a strain gauge, the component of the drill bit is a bottom holeassembly, the material is cement and the known mechanical rock propertyincludes values for C₁₁ and C₁₂.
 19. A method of obtaining stress andstrain comprising: obtaining, with a processor, acoustical signalsindicative of a force and a displacement, the signals obtained from oneor more sensors positioned on a component of a bottom hole assembly, theforce and displacement of a drill bit interacting with a material withknown mechanical rock properties; processing the forces and thedisplacements to obtain a set of scalars that conform to a stress strainrelationship of the material with known mechanical rock properties; andapplying the set of scalars to the forces and the displacements toobtain stress and strain generated by the drill bit interacting with thematerial.
 20. The method of claim 19 wherein the stress strainrelationship is: ${\begin{bmatrix}{Dd}_{1} & ( {{Ed}_{2} + {Fd}_{3}} ) \\{Ed}_{2} & ( {{Dd}_{1} + {Fd}_{3}} ) \\{Fd}_{3} & ( {{Dd}_{1} + {Ed}_{2}} )\end{bmatrix}\begin{bmatrix}C_{11} \\C_{12}\end{bmatrix}} = \begin{bmatrix}{Aa}_{1} \\{Ba}_{2} \\{Ca}_{3}\end{bmatrix}$ where: the a values are forces or accelerations acting onthe drill bit interacting with the sample; the d values aredisplacements the drill bit interacting with the sample; where C₁₁ andC₁₂ are populated with known rock properties of the sample; and A-F arethe at least one set of scalars being obtained.
 21. The method of claim20 wherein: the a values are an axial (a1) acceleration, a lateral (a2)acceleration and a centripetal (a3) acceleration of the drill bit; the dvalues are an axial (d1) displacement, a lateral (d2) displacement and acentripetal (d3) displacement; the obtained stresses are an axialstress, a lateral stress and a centripetal stress from the respectivescalars A, B, and C multiplied by the respective a-values a1, a2 and a3;and the obtained strains are an axial strain, a lateral strain and acentripetal strain from the respective scalars D, E, and F multiplied bythe respective d-values d1, d2, and d3.
 22. The method of claim 2further comprising: obtaining the scalars by: using a computer model ofthe stresses and strains of a sample material with known mechanical rockproperties to obtain the stresses generated on the sample by a drill bitinteracting with the sample and using the computer model to obtain thestrains experienced by the sample of the drill bit interacting with thesample, further to obtain the scalars that conform to a stress strainrelationship of the sample with the known mechanical computer model. 23.The method of claim 22 wherein the stress strain relationship is:${A = \frac{\sigma}{a_{1}}},{B = \frac{\tau}{a_{2}}},{C = \frac{\tau}{a_{3}}},{D = \frac{ɛ}{d_{1}}},{E = \frac{ɛ}{d_{2}}},{F = \frac{ɛ}{d_{3}}}$where: the a values are acceleration data of the drill bit interactingwith the formation with respect to an axis of drilling; the d values aredisplacement data of the drill bit interacting with the formation andwith respect to the axis of drilling; where σ and τ are the axial stressand one of lateral or angular stress are populated with the computermodel; ε are the axial strain and one of lateral or angular stress arepopulated with the computer model; and A-F are the scalars.
 24. Anapparatus comprising: a hardware processor in communication with anon-transitory computer readable media including computer executableinstructions to perform a method of: accessing a first set of acousticalsignals obtained from one or more sensors, the acoustical signalsgenerated from a drill bit interacting with a rock formation whiledrilling a wellbore; processing the first set of acoustical signals toobtain forces acting on the drill bit interacting with the rockformation while drilling the wellbore and to obtain displacements of thedrill bit interacting with the rock formation while drilling thewellbore; scaling the obtained forces acting on the drill bitinteracting with the rock formation while drilling the wellbore and theobtained displacements of the drill bit interacting with the rockformation while drilling the wellbore to obtain informationrepresentative of stresses and strains of the rock formation; andprocessing the scaled forces and the scaled displacements to obtain andstore in a tangible memory at least one set of data valuesrepresentative of a mechanical rock property of the rock formation alongthe wellbore created by the drill bit interacting with the rockformation for a period of time.
 25. The apparatus of claim 24 whereinthe computer executable instructions are further to perform the methodof: applying scalars to the obtained forces acting on the drill bitinteracting with the rock formation while drilling the wellbore and theobtained displacements of the drill bit interacting with the rockformation while drilling the wellbore, the scalars derived fromacoustical signals generated from the drill bit acting on a sample withknown mechanical rock properties; and obtaining the scalars by:receiving a second set of acoustical signals obtained from the one ormore sensors wherein the one or more sensors are positioned on acomponent of a bottom hole assembly, the second set of acousticalsignals generated from the drill bit interacting with the sample;processing the second set of acoustical signals to obtain forces actingon the drill bit interacting with the sample and to obtain displacementsof the drill bit interacting with the sample; processing the obtainedforces acting on the drill bit interacting with the sample and theobtained displacements of the drill bit interacting with the sample toobtain the scalars that conform to a stress strain relationship of thesample with known mechanical rock properties.
 26. The apparatus of claim25 wherein the stress strain relationship is: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are forces or accelerations acting onthe drill bit interacting with the sample; the d values aredisplacements the drill bit interacting with the sample; where C₁₁ andC₁₂ are populated with known rock properties of the sample; and A-F arethe at least one set of scalars being obtained.
 27. An apparatuscomprising: a hardware processor in communication with a non-transitorycomputer readable media including computer executable instructions toperform a method of: obtaining a force data and a displacement data fromsignals obtained from one or more sensors positioned on a component of adrilling tool proximate a drill bit, the force data and the displacementdata being of a drill bit interacting with a material with knownmechanical rock properties; processing the force data and thedisplacement data to obtain and store in a tangible memory a set ofscalars that conform to a stress strain relationship of the materialwith a known mechanical rock property.
 28. The apparatus of claim 27wherein the stress strain relationship is: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{12}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{12}} \\0 & 0 & {- a_{3}} & {d_{1}C_{12}} & {d_{2}C_{12}} & {d_{3}C_{11}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are the force data with respect to anaxis of material symmetry of the material; the d values are thedisplacement data with respect to the axis of material symmetry of thematerial; where C₁₁ and C₁₂ are populated with the known rock propertyof the material; and A-F are the set of scalars.
 29. The apparatus ofclaim 27 wherein the stress strain relationship is: ${\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{1}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = \begin{bmatrix}0 \\0 \\0\end{bmatrix}$ where: the a values are the force data with respect to anaxis of material symmetry of the material; the d values are thedisplacement data with respect to the axis of material symmetry of thematerial; where the Cijs are populated with the known mechanical rockproperty of the material; and A-F are the set of scalars.